id	sid	tid	token	lemma	pos
ap-7678	1	1	acta	acta	PROPN
ap-7678	1	2	polytechnica	polytechnica	PROPN
ap-7678	1	3	https://doi.org/10.14311/ap.2022.62.0063	https://doi.org/10.14311/ap.2022.62.0063	PROPN
ap-7678	1	4	acta	acta	PROPN
ap-7678	1	5	polytechnica	polytechnica	PROPN
ap-7678	1	6	62(1):63–79	62(1):63–79	ADP
ap-7678	1	7	,	,	PUNCT
ap-7678	1	8	2022	2022	NUM
ap-7678	1	9	©	©	ADP
ap-7678	1	10	2022	2022	NUM
ap-7678	1	11	the	the	DET
ap-7678	1	12	author(s	author(s	NOUN
ap-7678	1	13	)	)	PUNCT
ap-7678	1	14	.	.	PUNCT
ap-7678	2	1	licensed	license	VERB
ap-7678	2	2	under	under	ADP
ap-7678	2	3	a	a	DET
ap-7678	2	4	cc	cc	NOUN
ap-7678	2	5	-	-	PUNCT
ap-7678	2	6	by	by	ADP
ap-7678	2	7	4.0	4.0	NUM
ap-7678	2	8	licence	licence	NOUN
ap-7678	2	9	published	publish	VERB
ap-7678	2	10	by	by	ADP
ap-7678	2	11	the	the	DET
ap-7678	2	12	czech	czech	PROPN
ap-7678	2	13	technical	technical	PROPN
ap-7678	2	14	university	university	PROPN
ap-7678	2	15	in	in	ADP
ap-7678	2	16	prague	prague	PROPN
ap-7678	2	17	orthonormal	orthonormal	ADJ
ap-7678	2	18	polynomial	polynomial	ADJ
ap-7678	2	19	projection	projection	NOUN
ap-7678	2	20	quantization	quantization	NOUN
ap-7678	2	21	:	:	PUNCT
ap-7678	2	22	an	an	DET
ap-7678	2	23	algebraic	algebraic	ADJ
ap-7678	2	24	eigenenergy	eigenenergy	NOUN
ap-7678	2	25	bounding	bounding	NOUN
ap-7678	2	26	method	method	NOUN
ap-7678	2	27	carlos	carlos	PROPN
ap-7678	2	28	r.	r.	PROPN
ap-7678	2	29	handy	handy	PROPN
ap-7678	2	30	texas	texas	PROPN
ap-7678	2	31	southern	southern	PROPN
ap-7678	2	32	university	university	PROPN
ap-7678	2	33	,	,	PUNCT
ap-7678	2	34	3100	3100	NUM
ap-7678	2	35	cleburne	cleburne	PROPN
ap-7678	2	36	st	st	PROPN
ap-7678	2	37	.	.	PROPN
ap-7678	2	38	,	,	PUNCT
ap-7678	2	39	houston	houston	PROPN
ap-7678	2	40	,	,	PUNCT
ap-7678	2	41	texas	texas	PROPN
ap-7678	2	42	,	,	PUNCT
ap-7678	2	43	u.s.a	u.s.a	PROPN
ap-7678	2	44	.	.	PROPN
ap-7678	2	45	correspondence	correspondence	NOUN
ap-7678	2	46	:	:	PUNCT
ap-7678	2	47	carlos.handy@tsu.edu	carlos.handy@tsu.edu	NOUN
ap-7678	2	48	abstract	abstract	NOUN
ap-7678	2	49	.	.	PUNCT
ap-7678	3	1	the	the	DET
ap-7678	3	2	ability	ability	NOUN
ap-7678	3	3	to	to	PART
ap-7678	3	4	generate	generate	VERB
ap-7678	3	5	tight	tight	ADJ
ap-7678	3	6	eigenenergy	eigenenergy	NOUN
ap-7678	3	7	bounds	bound	NOUN
ap-7678	3	8	for	for	ADP
ap-7678	3	9	low	low	ADJ
ap-7678	3	10	dimension	dimension	NOUN
ap-7678	3	11	bosonic	bosonic	NOUN
ap-7678	3	12	or	or	CCONJ
ap-7678	3	13	ferminonic	ferminonic	ADJ
ap-7678	3	14	,	,	PUNCT
ap-7678	3	15	hermitian	hermitian	ADJ
ap-7678	3	16	or	or	CCONJ
ap-7678	3	17	non	non	ADJ
ap-7678	3	18	-	-	ADJ
ap-7678	3	19	hermitian	hermitian	ADJ
ap-7678	3	20	,	,	PUNCT
ap-7678	3	21	schrödinger	schrödinger	ADJ
ap-7678	3	22	operator	operator	NOUN
ap-7678	3	23	problems	problem	NOUN
ap-7678	3	24	is	be	AUX
ap-7678	3	25	an	an	DET
ap-7678	3	26	important	important	ADJ
ap-7678	3	27	objective	objective	NOUN
ap-7678	3	28	in	in	ADP
ap-7678	3	29	the	the	DET
ap-7678	3	30	computation	computation	NOUN
ap-7678	3	31	of	of	ADP
ap-7678	3	32	quantum	quantum	NOUN
ap-7678	3	33	systems	system	NOUN
ap-7678	3	34	.	.	PUNCT
ap-7678	4	1	very	very	ADV
ap-7678	4	2	few	few	ADJ
ap-7678	4	3	methods	method	NOUN
ap-7678	4	4	can	can	AUX
ap-7678	4	5	simultaneously	simultaneously	ADV
ap-7678	4	6	generate	generate	VERB
ap-7678	4	7	lower	low	ADJ
ap-7678	4	8	and	and	CCONJ
ap-7678	4	9	upper	upper	ADJ
ap-7678	4	10	bounds	bound	NOUN
ap-7678	4	11	.	.	PUNCT
ap-7678	5	1	one	one	NUM
ap-7678	5	2	of	of	ADP
ap-7678	5	3	these	these	PRON
ap-7678	5	4	is	be	AUX
ap-7678	5	5	the	the	DET
ap-7678	5	6	eigenvalue	eigenvalue	PROPN
ap-7678	5	7	moment	moment	NOUN
ap-7678	5	8	method	method	NOUN
ap-7678	5	9	(	(	PUNCT
ap-7678	5	10	emm	emm	PROPN
ap-7678	5	11	)	)	PUNCT
ap-7678	5	12	originally	originally	ADV
ap-7678	5	13	introduced	introduce	VERB
ap-7678	5	14	by	by	ADP
ap-7678	5	15	handy	handy	ADJ
ap-7678	5	16	and	and	CCONJ
ap-7678	5	17	besssis	besssis	NOUN
ap-7678	5	18	,	,	PUNCT
ap-7678	5	19	exploiting	exploit	VERB
ap-7678	5	20	the	the	DET
ap-7678	5	21	use	use	NOUN
ap-7678	5	22	of	of	ADP
ap-7678	5	23	semidefinite	semidefinite	PROPN
ap-7678	5	24	programming	programming	PROPN
ap-7678	5	25	/	/	SYM
ap-7678	5	26	nonlinear	nonlinear	ADJ
ap-7678	5	27	-	-	PUNCT
ap-7678	5	28	convex	convex	ADJ
ap-7678	5	29	optimization	optimization	NOUN
ap-7678	5	30	(	(	PUNCT
ap-7678	5	31	sdp	sdp	NOUN
ap-7678	5	32	)	)	PUNCT
ap-7678	5	33	techniques	technique	NOUN
ap-7678	5	34	as	as	SCONJ
ap-7678	5	35	applied	apply	VERB
ap-7678	5	36	to	to	ADP
ap-7678	5	37	the	the	DET
ap-7678	5	38	positivity	positivity	NOUN
ap-7678	5	39	properties	property	NOUN
ap-7678	5	40	of	of	ADP
ap-7678	5	41	the	the	DET
ap-7678	5	42	multidimensional	multidimensional	ADJ
ap-7678	5	43	bosonic	bosonic	NOUN
ap-7678	5	44	ground	ground	NOUN
ap-7678	5	45	state	state	NOUN
ap-7678	5	46	for	for	ADP
ap-7678	5	47	a	a	DET
ap-7678	5	48	large	large	ADJ
ap-7678	5	49	class	class	NOUN
ap-7678	5	50	of	of	ADP
ap-7678	5	51	important	important	ADJ
ap-7678	5	52	physical	physical	ADJ
ap-7678	5	53	systems	system	NOUN
ap-7678	5	54	(	(	PUNCT
ap-7678	5	55	i.e.	i.e.	X
ap-7678	5	56	those	those	PRON
ap-7678	5	57	admitting	admit	VERB
ap-7678	5	58	a	a	DET
ap-7678	5	59	moments	moment	NOUN
ap-7678	5	60	’	'	PUNCT
ap-7678	5	61	representation	representation	NOUN
ap-7678	5	62	)	)	PUNCT
ap-7678	5	63	.	.	PUNCT
ap-7678	6	1	a	a	DET
ap-7678	6	2	recent	recent	ADJ
ap-7678	6	3	breakthrough	breakthrough	NOUN
ap-7678	6	4	has	have	AUX
ap-7678	6	5	been	be	AUX
ap-7678	6	6	achieved	achieve	VERB
ap-7678	6	7	through	through	ADP
ap-7678	6	8	another	another	PRON
ap-7678	6	9	,	,	PUNCT
ap-7678	6	10	simpler	simple	ADJ
ap-7678	6	11	,	,	PUNCT
ap-7678	6	12	moment	moment	NOUN
ap-7678	6	13	representation	representation	NOUN
ap-7678	6	14	based	base	VERB
ap-7678	6	15	quantization	quantization	NOUN
ap-7678	6	16	formalism	formalism	NOUN
ap-7678	6	17	,	,	PUNCT
ap-7678	6	18	the	the	DET
ap-7678	6	19	orthonormal	orthonormal	ADJ
ap-7678	6	20	polynomial	polynomial	ADJ
ap-7678	6	21	projection	projection	NOUN
ap-7678	6	22	quantization	quantization	NOUN
ap-7678	6	23	bounding	bounding	NOUN
ap-7678	6	24	method	method	NOUN
ap-7678	6	25	(	(	PUNCT
ap-7678	6	26	oppq	oppq	PROPN
ap-7678	6	27	-	-	PUNCT
ap-7678	6	28	bm	bm	PROPN
ap-7678	6	29	)	)	PUNCT
ap-7678	6	30	.	.	PUNCT
ap-7678	7	1	it	it	PRON
ap-7678	7	2	is	be	AUX
ap-7678	7	3	purely	purely	ADV
ap-7678	7	4	algebraic	algebraic	ADJ
ap-7678	7	5	and	and	CCONJ
ap-7678	7	6	does	do	AUX
ap-7678	7	7	not	not	PART
ap-7678	7	8	require	require	VERB
ap-7678	7	9	any	any	DET
ap-7678	7	10	sdp	sdp	NOUN
ap-7678	7	11	analysis	analysis	NOUN
ap-7678	7	12	.	.	PUNCT
ap-7678	8	1	we	we	PRON
ap-7678	8	2	discuss	discuss	VERB
ap-7678	8	3	its	its	PRON
ap-7678	8	4	essential	essential	ADJ
ap-7678	8	5	structure	structure	NOUN
ap-7678	8	6	in	in	ADP
ap-7678	8	7	the	the	DET
ap-7678	8	8	context	context	NOUN
ap-7678	8	9	of	of	ADP
ap-7678	8	10	several	several	ADJ
ap-7678	8	11	one	one	NUM
ap-7678	8	12	dimensional	dimensional	ADJ
ap-7678	8	13	examples	example	NOUN
ap-7678	8	14	.	.	PUNCT
ap-7678	9	1	keywords	keyword	NOUN
ap-7678	9	2	:	:	PUNCT
ap-7678	9	3	eigenvalue	eigenvalue	ADJ
ap-7678	9	4	bounding	bounding	NOUN
ap-7678	9	5	methods	method	NOUN
ap-7678	9	6	,	,	PUNCT
ap-7678	9	7	hermitian	hermitian	ADJ
ap-7678	9	8	and	and	CCONJ
ap-7678	9	9	non	non	ADJ
ap-7678	9	10	-	-	ADJ
ap-7678	9	11	hermitian	hermitian	ADJ
ap-7678	9	12	linear	linear	PROPN
ap-7678	9	13	operators	operator	NOUN
ap-7678	9	14	.	.	PUNCT
ap-7678	10	1	1	1	X
ap-7678	10	2	.	.	X
ap-7678	10	3	introduction	introduction	NOUN
ap-7678	10	4	the	the	DET
ap-7678	10	5	eigenvalue	eigenvalue	ADJ
ap-7678	10	6	bounding	bounding	NOUN
ap-7678	10	7	problem	problem	NOUN
ap-7678	10	8	for	for	ADP
ap-7678	10	9	linear	linear	ADJ
ap-7678	10	10	ordinary	ordinary	ADJ
ap-7678	10	11	differential	differential	ADJ
ap-7678	10	12	equations	equation	NOUN
ap-7678	10	13	,	,	PUNCT
ap-7678	10	14	or	or	CCONJ
ap-7678	10	15	linear	linear	VERB
ap-7678	10	16	partial	partial	ADJ
ap-7678	10	17	differential	differential	NOUN
ap-7678	10	18	equations	equation	NOUN
ap-7678	10	19	(	(	PUNCT
ap-7678	10	20	i.e.	i.e.	X
ap-7678	10	21	lode	lode	ADJ
ap-7678	10	22	/	/	SYM
ap-7678	10	23	lpde	lpde	NOUN
ap-7678	10	24	)	)	PUNCT
ap-7678	10	25	has	have	AUX
ap-7678	10	26	been	be	AUX
ap-7678	10	27	an	an	DET
ap-7678	10	28	active	active	ADJ
ap-7678	10	29	area	area	NOUN
ap-7678	10	30	of	of	ADP
ap-7678	10	31	research	research	NOUN
ap-7678	10	32	for	for	ADP
ap-7678	10	33	many	many	ADJ
ap-7678	10	34	decades	decade	NOUN
ap-7678	10	35	.	.	PUNCT
ap-7678	11	1	in	in	ADP
ap-7678	11	2	the	the	DET
ap-7678	11	3	context	context	NOUN
ap-7678	11	4	of	of	ADP
ap-7678	11	5	quantum	quantum	ADJ
ap-7678	11	6	physical	physical	ADJ
ap-7678	11	7	systems	system	NOUN
ap-7678	11	8	,	,	PUNCT
ap-7678	11	9	as	as	SCONJ
ap-7678	11	10	represented	represent	VERB
ap-7678	11	11	by	by	ADP
ap-7678	11	12	the	the	DET
ap-7678	11	13	multidimensional	multidimensional	ADJ
ap-7678	11	14	schrödinger	schrödinger	ADJ
ap-7678	11	15	equation	equation	NOUN
ap-7678	11	16	,	,	PUNCT
ap-7678	11	17	−	−	PROPN
ap-7678	11	18	ℏ2	ℏ2	NOUN
ap-7678	11	19	2	2	NUM
ap-7678	11	20	m	m	VERB
ap-7678	11	21	∇2ψ(−→r	∇2ψ(−→r	PRON
ap-7678	11	22	)	)	PUNCT
ap-7678	12	1	+	+	CCONJ
ap-7678	12	2	v	v	X
ap-7678	12	3	(	(	PUNCT
ap-7678	12	4	−→r	−→r	NOUN
ap-7678	12	5	)	)	PUNCT
ap-7678	12	6	ψ(−→r	ψ(−→r	PROPN
ap-7678	12	7	)	)	PUNCT
ap-7678	13	1	=	=	SYM
ap-7678	13	2	eψ(−→r	eψ(−→r	PROPN
ap-7678	13	3	)	)	PUNCT
ap-7678	13	4	,	,	PUNCT
ap-7678	13	5	(	(	PUNCT
ap-7678	13	6	1	1	X
ap-7678	13	7	)	)	PUNCT
ap-7678	13	8	the	the	DET
ap-7678	13	9	generation	generation	NOUN
ap-7678	13	10	of	of	ADP
ap-7678	13	11	upper	upper	ADJ
ap-7678	13	12	bounds	bound	NOUN
ap-7678	13	13	to	to	ADP
ap-7678	13	14	the	the	DET
ap-7678	13	15	individual	individual	ADJ
ap-7678	13	16	discrete	discrete	ADJ
ap-7678	13	17	state	state	NOUN
ap-7678	13	18	energies	energy	NOUN
ap-7678	13	19	is	be	AUX
ap-7678	13	20	readily	readily	ADV
ap-7678	13	21	obtainable	obtainable	ADJ
ap-7678	13	22	through	through	ADP
ap-7678	13	23	such	such	ADJ
ap-7678	13	24	well	well	ADV
ap-7678	13	25	known	know	VERB
ap-7678	13	26	methods	method	NOUN
ap-7678	13	27	as	as	ADP
ap-7678	13	28	that	that	PRON
ap-7678	13	29	of	of	ADP
ap-7678	13	30	rayleigh	rayleigh	PROPN
ap-7678	13	31	-	-	PUNCT
ap-7678	13	32	ritz	ritz	PROPN
ap-7678	13	33	(	(	PUNCT
ap-7678	13	34	rr	rr	NOUN
ap-7678	13	35	)	)	PUNCT
ap-7678	14	1	[	[	X
ap-7678	14	2	1	1	NUM
ap-7678	14	3	]	]	PUNCT
ap-7678	14	4	.	.	PUNCT
ap-7678	15	1	the	the	DET
ap-7678	15	2	challenge	challenge	NOUN
ap-7678	15	3	has	have	AUX
ap-7678	15	4	been	be	AUX
ap-7678	15	5	to	to	PART
ap-7678	15	6	find	find	VERB
ap-7678	15	7	an	an	DET
ap-7678	15	8	equally	equally	ADV
ap-7678	15	9	effective	effective	ADJ
ap-7678	15	10	lower	low	ADJ
ap-7678	15	11	bound	bind	VERB
ap-7678	15	12	method	method	NOUN
ap-7678	15	13	.	.	PUNCT
ap-7678	16	1	a	a	DET
ap-7678	16	2	well	well	ADV
ap-7678	16	3	known	know	VERB
ap-7678	16	4	lower	lower	ADV
ap-7678	16	5	bound	bind	VERB
ap-7678	16	6	method	method	NOUN
ap-7678	16	7	is	be	AUX
ap-7678	16	8	that	that	SCONJ
ap-7678	16	9	associated	associate	VERB
ap-7678	16	10	with	with	ADP
ap-7678	16	11	temple	temple	NOUN
ap-7678	16	12	[	[	X
ap-7678	16	13	2	2	NUM
ap-7678	16	14	]	]	PUNCT
ap-7678	16	15	;	;	PUNCT
ap-7678	16	16	however	however	ADV
ap-7678	16	17	,	,	PUNCT
ap-7678	16	18	its	its	PRON
ap-7678	16	19	convergence	convergence	NOUN
ap-7678	16	20	rate	rate	NOUN
ap-7678	16	21	is	be	AUX
ap-7678	16	22	slow	slow	ADJ
ap-7678	16	23	.	.	PUNCT
ap-7678	17	1	nevertheless	nevertheless	ADV
ap-7678	17	2	,	,	PUNCT
ap-7678	17	3	it	it	PRON
ap-7678	17	4	has	have	AUX
ap-7678	17	5	served	serve	VERB
ap-7678	17	6	as	as	ADP
ap-7678	17	7	a	a	DET
ap-7678	17	8	spring	spring	NOUN
ap-7678	17	9	board	board	NOUN
ap-7678	17	10	for	for	ADP
ap-7678	17	11	other	other	ADJ
ap-7678	17	12	more	more	ADV
ap-7678	17	13	effective	effective	ADJ
ap-7678	17	14	lower	low	ADJ
ap-7678	17	15	bound	bind	VERB
ap-7678	17	16	formulations	formulation	NOUN
ap-7678	17	17	.	.	PUNCT
ap-7678	18	1	among	among	ADP
ap-7678	18	2	these	these	PRON
ap-7678	18	3	is	be	AUX
ap-7678	18	4	the	the	DET
ap-7678	18	5	work	work	NOUN
ap-7678	18	6	by	by	ADP
ap-7678	18	7	marmorino	marmorino	PROPN
ap-7678	18	8	et	et	PROPN
ap-7678	18	9	al	al	PROPN
ap-7678	18	10	.	.	PUNCT
ap-7678	19	1	[	[	X
ap-7678	19	2	3	3	NUM
ap-7678	19	3	]	]	PUNCT
ap-7678	19	4	,	,	PUNCT
ap-7678	19	5	and	and	CCONJ
ap-7678	19	6	more	more	ADV
ap-7678	19	7	recently	recently	ADV
ap-7678	19	8	,	,	PUNCT
ap-7678	19	9	that	that	PRON
ap-7678	19	10	of	of	ADP
ap-7678	19	11	martinazzo	martinazzo	NOUN
ap-7678	19	12	and	and	CCONJ
ap-7678	19	13	pollak	pollak	PROPN
ap-7678	20	1	[	[	X
ap-7678	20	2	4	4	NUM
ap-7678	20	3	]	]	PUNCT
ap-7678	20	4	.	.	PUNCT
ap-7678	21	1	the	the	DET
ap-7678	21	2	latter	latter	ADJ
ap-7678	21	3	are	be	AUX
ap-7678	21	4	able	able	ADJ
ap-7678	21	5	to	to	PART
ap-7678	21	6	improve	improve	VERB
ap-7678	21	7	upon	upon	SCONJ
ap-7678	21	8	the	the	DET
ap-7678	21	9	convergence	convergence	NOUN
ap-7678	21	10	rate	rate	NOUN
ap-7678	21	11	of	of	ADP
ap-7678	21	12	temple	temple	PROPN
ap-7678	21	13	’s	’s	PART
ap-7678	21	14	lower	low	ADJ
ap-7678	21	15	bound	bind	VERB
ap-7678	21	16	formulation	formulation	NOUN
ap-7678	21	17	.	.	PUNCT
ap-7678	22	1	despite	despite	SCONJ
ap-7678	22	2	these	these	DET
ap-7678	22	3	successes	success	NOUN
ap-7678	22	4	,	,	PUNCT
ap-7678	22	5	two	two	NUM
ap-7678	22	6	important	important	ADJ
ap-7678	22	7	facts	fact	NOUN
ap-7678	22	8	remain	remain	VERB
ap-7678	22	9	.	.	PUNCT
ap-7678	23	1	the	the	DET
ap-7678	23	2	first	first	ADJ
ap-7678	23	3	is	be	AUX
ap-7678	23	4	that	that	SCONJ
ap-7678	23	5	all	all	DET
ap-7678	23	6	the	the	DET
ap-7678	23	7	above	above	ADJ
ap-7678	23	8	bounding	bounding	NOUN
ap-7678	23	9	methods	method	NOUN
ap-7678	23	10	require	require	VERB
ap-7678	23	11	the	the	DET
ap-7678	23	12	use	use	NOUN
ap-7678	23	13	of	of	ADP
ap-7678	23	14	two	two	NUM
ap-7678	23	15	different	different	ADJ
ap-7678	23	16	bounding	bounding	NOUN
ap-7678	23	17	formulations	formulation	NOUN
ap-7678	23	18	.	.	PUNCT
ap-7678	24	1	one	one	NUM
ap-7678	24	2	for	for	ADP
ap-7678	24	3	the	the	DET
ap-7678	24	4	upper	upper	ADJ
ap-7678	24	5	bounds	bound	NOUN
ap-7678	24	6	(	(	PUNCT
ap-7678	24	7	rr	rr	NOUN
ap-7678	24	8	)	)	PUNCT
ap-7678	24	9	,	,	PUNCT
ap-7678	24	10	another	another	PRON
ap-7678	24	11	for	for	ADP
ap-7678	24	12	the	the	DET
ap-7678	24	13	lower	low	ADJ
ap-7678	24	14	bounds	bound	NOUN
ap-7678	24	15	.	.	PUNCT
ap-7678	25	1	that	that	PRON
ap-7678	25	2	is	is	ADV
ap-7678	25	3	,	,	PUNCT
ap-7678	25	4	they	they	PRON
ap-7678	25	5	do	do	AUX
ap-7678	25	6	not	not	PART
ap-7678	25	7	define	define	VERB
ap-7678	25	8	a	a	DET
ap-7678	25	9	unified	unified	ADJ
ap-7678	25	10	theoretical	theoretical	ADJ
ap-7678	25	11	framework	framework	NOUN
ap-7678	25	12	for	for	ADP
ap-7678	25	13	simultaneously	simultaneously	ADV
ap-7678	25	14	generating	generate	VERB
ap-7678	25	15	lower	low	ADJ
ap-7678	25	16	and	and	CCONJ
ap-7678	25	17	upper	upper	ADJ
ap-7678	25	18	bounds	bound	NOUN
ap-7678	25	19	.	.	PUNCT
ap-7678	26	1	additionally	additionally	ADV
ap-7678	26	2	,	,	PUNCT
ap-7678	26	3	these	these	DET
ap-7678	26	4	methods	method	NOUN
ap-7678	26	5	are	be	AUX
ap-7678	26	6	based	base	VERB
ap-7678	26	7	on	on	ADP
ap-7678	26	8	a	a	DET
ap-7678	26	9	hilbert	hilbert	NOUN
ap-7678	26	10	space	space	NOUN
ap-7678	26	11	representation	representation	NOUN
ap-7678	26	12	for	for	ADP
ap-7678	26	13	quantum	quantum	NOUN
ap-7678	26	14	systems	system	NOUN
ap-7678	26	15	,	,	PUNCT
ap-7678	26	16	dependent	dependent	ADJ
ap-7678	26	17	on	on	ADP
ap-7678	26	18	the	the	DET
ap-7678	26	19	existence	existence	NOUN
ap-7678	26	20	of	of	ADP
ap-7678	26	21	hermitian	hermitian	ADJ
ap-7678	26	22	hamiltonians	hamiltonian	NOUN
ap-7678	26	23	.	.	PUNCT
ap-7678	27	1	they	they	PRON
ap-7678	27	2	are	be	AUX
ap-7678	27	3	of	of	ADP
ap-7678	27	4	little	little	ADJ
ap-7678	27	5	relevance	relevance	NOUN
ap-7678	27	6	for	for	ADP
ap-7678	27	7	bounding	bound	VERB
ap-7678	27	8	the	the	DET
ap-7678	27	9	real	real	ADJ
ap-7678	27	10	/	/	SYM
ap-7678	27	11	complex	complex	ADJ
ap-7678	27	12	eigenenergies	eigenenergie	NOUN
ap-7678	27	13	of	of	ADP
ap-7678	27	14	nonhermitian	nonhermitian	ADJ
ap-7678	27	15	systems	system	NOUN
ap-7678	27	16	,	,	PUNCT
ap-7678	27	17	particularly	particularly	ADV
ap-7678	27	18	those	those	PRON
ap-7678	27	19	corresponding	correspond	VERB
ap-7678	27	20	to	to	ADP
ap-7678	27	21	pt	pt	NOUN
ap-7678	27	22	-	-	PUNCT
ap-7678	27	23	symmetry	symmetry	NOUN
ap-7678	27	24	breaking	break	VERB
ap-7678	27	25	systems	system	NOUN
ap-7678	27	26	[	[	X
ap-7678	27	27	5–8	5–8	NOUN
ap-7678	27	28	]	]	PUNCT
ap-7678	27	29	.	.	PUNCT
ap-7678	28	1	in	in	ADP
ap-7678	28	2	this	this	DET
ap-7678	28	3	work	work	NOUN
ap-7678	28	4	we	we	PRON
ap-7678	28	5	present	present	VERB
ap-7678	28	6	a	a	DET
ap-7678	28	7	novel	novel	ADJ
ap-7678	28	8	approach	approach	NOUN
ap-7678	28	9	that	that	PRON
ap-7678	28	10	can	can	AUX
ap-7678	28	11	generate	generate	VERB
ap-7678	28	12	tight	tight	ADJ
ap-7678	28	13	bounds	bound	NOUN
ap-7678	28	14	for	for	ADP
ap-7678	28	15	the	the	DET
ap-7678	28	16	discrete	discrete	ADJ
ap-7678	28	17	states	state	NOUN
ap-7678	28	18	of	of	ADP
ap-7678	28	19	bosonic	bosonic	NOUN
ap-7678	28	20	or	or	CCONJ
ap-7678	28	21	fermionic	fermionic	NOUN
ap-7678	28	22	,	,	PUNCT
ap-7678	28	23	low	low	ADJ
ap-7678	28	24	dimension	dimension	NOUN
ap-7678	28	25	,	,	PUNCT
ap-7678	28	26	systems	system	NOUN
ap-7678	28	27	,	,	PUNCT
ap-7678	28	28	regardless	regardless	ADV
ap-7678	28	29	if	if	SCONJ
ap-7678	28	30	they	they	PRON
ap-7678	28	31	are	be	AUX
ap-7678	28	32	hermitian	hermitian	ADJ
ap-7678	28	33	or	or	CCONJ
ap-7678	28	34	not	not	PART
ap-7678	28	35	.	.	PUNCT
ap-7678	29	1	it	it	PRON
ap-7678	29	2	is	be	AUX
ap-7678	29	3	referred	refer	VERB
ap-7678	29	4	to	to	ADP
ap-7678	29	5	as	as	ADP
ap-7678	29	6	the	the	DET
ap-7678	29	7	orthonormal	orthonormal	ADJ
ap-7678	29	8	polynomial	polynomial	ADJ
ap-7678	29	9	projection	projection	NOUN
ap-7678	29	10	quantization	quantization	NOUN
ap-7678	29	11	bounding	bounding	NOUN
ap-7678	29	12	method	method	NOUN
ap-7678	29	13	(	(	PUNCT
ap-7678	29	14	oppq	oppq	PROPN
ap-7678	29	15	-	-	PUNCT
ap-7678	29	16	bm	bm	PROPN
ap-7678	29	17	)	)	PUNCT
ap-7678	29	18	,	,	PUNCT
ap-7678	29	19	as	as	SCONJ
ap-7678	29	20	developed	develop	VERB
ap-7678	29	21	by	by	ADP
ap-7678	29	22	handy	handy	ADJ
ap-7678	29	23	[	[	X
ap-7678	29	24	9	9	NUM
ap-7678	29	25	]	]	PUNCT
ap-7678	29	26	;	;	PUNCT
ap-7678	29	27	and	and	CCONJ
ap-7678	29	28	based	base	VERB
ap-7678	29	29	,	,	PUNCT
ap-7678	29	30	in	in	ADP
ap-7678	29	31	part	part	NOUN
ap-7678	29	32	,	,	PUNCT
ap-7678	29	33	on	on	ADP
ap-7678	29	34	a	a	DET
ap-7678	29	35	related	relate	VERB
ap-7678	29	36	method	method	NOUN
ap-7678	29	37	,	,	PUNCT
ap-7678	29	38	the	the	DET
ap-7678	29	39	oppq	oppq	NOUN
ap-7678	29	40	-	-	PUNCT
ap-7678	29	41	approximation	approximation	NOUN
ap-7678	29	42	method	method	NOUN
ap-7678	29	43	by	by	ADP
ap-7678	29	44	handy	handy	ADJ
ap-7678	29	45	and	and	CCONJ
ap-7678	29	46	vrinceanu	vrinceanu	NOUN
ap-7678	30	1	[	[	X
ap-7678	30	2	10	10	NUM
ap-7678	30	3	,	,	PUNCT
ap-7678	30	4	11	11	NUM
ap-7678	30	5	]	]	PUNCT
ap-7678	30	6	.	.	PUNCT
ap-7678	31	1	its	its	PRON
ap-7678	31	2	general	general	ADJ
ap-7678	31	3	structure	structure	NOUN
ap-7678	31	4	is	be	AUX
ap-7678	31	5	outlined	outline	VERB
ap-7678	31	6	in	in	ADP
ap-7678	31	7	the	the	DET
ap-7678	31	8	following	follow	VERB
ap-7678	31	9	sections	section	NOUN
ap-7678	31	10	,	,	PUNCT
ap-7678	31	11	through	through	ADP
ap-7678	31	12	representative	representative	ADJ
ap-7678	31	13	one	one	NUM
ap-7678	31	14	dimensional	dimensional	ADJ
ap-7678	31	15	systems	system	NOUN
ap-7678	31	16	,	,	PUNCT
ap-7678	31	17	both	both	CCONJ
ap-7678	31	18	hermitian	hermitian	ADJ
ap-7678	31	19	and	and	CCONJ
ap-7678	31	20	non	non	ADJ
ap-7678	31	21	-	-	ADJ
ap-7678	31	22	hermitian	hermitian	ADJ
ap-7678	31	23	.	.	PUNCT
ap-7678	32	1	we	we	PRON
ap-7678	32	2	outline	outline	VERB
ap-7678	32	3	the	the	DET
ap-7678	32	4	full	full	ADJ
ap-7678	32	5	oppq	oppq	PROPN
ap-7678	32	6	-	-	PUNCT
ap-7678	32	7	bm	bm	PROPN
ap-7678	32	8	theory	theory	NOUN
ap-7678	32	9	within	within	ADP
ap-7678	32	10	the	the	DET
ap-7678	32	11	context	context	NOUN
ap-7678	32	12	of	of	ADP
ap-7678	32	13	the	the	DET
ap-7678	32	14	one	one	NUM
ap-7678	32	15	dimensional	dimensional	ADJ
ap-7678	32	16	,	,	PUNCT
ap-7678	32	17	double	double	ADJ
ap-7678	32	18	well	well	ADV
ap-7678	32	19	,	,	PUNCT
ap-7678	32	20	sextic	sextic	ADJ
ap-7678	32	21	anharmonic	anharmonic	ADJ
ap-7678	32	22	oscillator	oscillator	NOUN
ap-7678	32	23	;	;	PUNCT
ap-7678	32	24	and	and	CCONJ
ap-7678	32	25	then	then	ADV
ap-7678	32	26	demonstrate	demonstrate	VERB
ap-7678	32	27	the	the	DET
ap-7678	32	28	existence	existence	NOUN
ap-7678	32	29	of	of	ADP
ap-7678	32	30	the	the	DET
ap-7678	32	31	key	key	ADJ
ap-7678	32	32	structures	structure	NOUN
ap-7678	32	33	necessary	necessary	ADJ
ap-7678	32	34	for	for	ADP
ap-7678	32	35	its	its	PRON
ap-7678	32	36	implementation	implementation	NOUN
ap-7678	32	37	to	to	ADP
ap-7678	32	38	the	the	DET
ap-7678	32	39	pt	pt	PROPN
ap-7678	32	40	symmetry	symmetry	NOUN
ap-7678	32	41	breaking	break	VERB
ap-7678	32	42	problem	problem	NOUN
ap-7678	32	43	with	with	ADP
ap-7678	32	44	potential	potential	ADJ
ap-7678	32	45	v	v	NOUN
ap-7678	32	46	(	(	PUNCT
ap-7678	32	47	x	x	NOUN
ap-7678	32	48	)	)	PUNCT
ap-7678	32	49	=	=	SYM
ap-7678	32	50	ix3	ix3	PROPN
ap-7678	32	51	+	+	CCONJ
ap-7678	32	52	iax	iax	PROPN
ap-7678	32	53	.	.	PUNCT
ap-7678	33	1	beyond	beyond	ADP
ap-7678	33	2	the	the	DET
ap-7678	33	3	theoretical	theoretical	ADJ
ap-7678	33	4	interest	interest	NOUN
ap-7678	33	5	in	in	ADP
ap-7678	33	6	bounds	bound	NOUN
ap-7678	33	7	,	,	PUNCT
ap-7678	33	8	they	they	PRON
ap-7678	33	9	are	be	AUX
ap-7678	33	10	also	also	ADV
ap-7678	33	11	of	of	ADP
ap-7678	33	12	practical	practical	ADJ
ap-7678	33	13	importance	importance	NOUN
ap-7678	33	14	for	for	ADP
ap-7678	33	15	delicate	delicate	ADJ
ap-7678	33	16	systems	system	NOUN
ap-7678	33	17	where	where	SCONJ
ap-7678	33	18	conventional	conventional	ADJ
ap-7678	33	19	computational	computational	ADJ
ap-7678	33	20	methods	method	NOUN
ap-7678	33	21	may	may	AUX
ap-7678	33	22	yield	yield	VERB
ap-7678	33	23	widely	widely	ADV
ap-7678	33	24	varying	vary	VERB
ap-7678	33	25	results	result	NOUN
ap-7678	33	26	.	.	PUNCT
ap-7678	34	1	that	that	PRON
ap-7678	34	2	is	is	ADV
ap-7678	34	3	,	,	PUNCT
ap-7678	34	4	the	the	DET
ap-7678	34	5	availability	availability	NOUN
ap-7678	34	6	of	of	ADP
ap-7678	34	7	tight	tight	ADJ
ap-7678	34	8	bounds	bound	NOUN
ap-7678	34	9	allows	allow	VERB
ap-7678	34	10	one	one	NUM
ap-7678	34	11	to	to	PART
ap-7678	34	12	discriminate	discriminate	VERB
ap-7678	34	13	between	between	ADP
ap-7678	34	14	competing	compete	VERB
ap-7678	34	15	theories	theory	NOUN
ap-7678	34	16	.	.	PUNCT
ap-7678	35	1	one	one	NUM
ap-7678	35	2	famous	famous	ADJ
ap-7678	35	3	problem	problem	NOUN
ap-7678	35	4	of	of	ADP
ap-7678	35	5	this	this	DET
ap-7678	35	6	type	type	NOUN
ap-7678	35	7	corresponds	correspond	VERB
ap-7678	35	8	to	to	ADP
ap-7678	35	9	the	the	DET
ap-7678	35	10	quadratic	quadratic	ADJ
ap-7678	35	11	zeeman	zeeman	NOUN
ap-7678	35	12	(	(	PUNCT
ap-7678	35	13	qzm	qzm	PROPN
ap-7678	35	14	)	)	PUNCT
ap-7678	35	15	effect	effect	NOUN
ap-7678	35	16	for	for	ADP
ap-7678	35	17	superstrong	superstrong	ADJ
ap-7678	35	18	magnetic	magnetic	ADJ
ap-7678	35	19	fields	field	NOUN
ap-7678	35	20	.	.	PUNCT
ap-7678	36	1	this	this	DET
ap-7678	36	2	problem	problem	NOUN
ap-7678	36	3	was	be	AUX
ap-7678	36	4	analyzed	analyze	VERB
ap-7678	36	5	through	through	ADP
ap-7678	36	6	many	many	ADJ
ap-7678	36	7	different	different	ADJ
ap-7678	36	8	types	type	NOUN
ap-7678	36	9	of	of	ADP
ap-7678	36	10	computational	computational	ADJ
ap-7678	36	11	methods	method	NOUN
ap-7678	36	12	,	,	PUNCT
ap-7678	36	13	resulting	result	VERB
ap-7678	36	14	in	in	ADP
ap-7678	36	15	a	a	DET
ap-7678	36	16	wide	wide	ADJ
ap-7678	36	17	range	range	NOUN
ap-7678	36	18	of	of	ADP
ap-7678	36	19	values	value	NOUN
ap-7678	36	20	for	for	ADP
ap-7678	36	21	the	the	DET
ap-7678	36	22	most	most	ADV
ap-7678	36	23	challenging	challenging	ADJ
ap-7678	36	24	state	state	NOUN
ap-7678	36	25	to	to	PART
ap-7678	36	26	compute	compute	VERB
ap-7678	36	27	:	:	PUNCT
ap-7678	36	28	the	the	DET
ap-7678	36	29	ground	ground	NOUN
ap-7678	36	30	state	state	NOUN
ap-7678	36	31	binding	bind	VERB
ap-7678	36	32	energy	energy	NOUN
ap-7678	36	33	.	.	PUNCT
ap-7678	37	1	this	this	PRON
ap-7678	37	2	was	be	AUX
ap-7678	37	3	reviewed	review	VERB
ap-7678	37	4	by	by	ADP
ap-7678	37	5	le	le	X
ap-7678	37	6	guillou	guillou	PROPN
ap-7678	37	7	and	and	CCONJ
ap-7678	37	8	zinn	zinn	PROPN
ap-7678	37	9	-	-	PUNCT
ap-7678	37	10	justin	justin	PROPN
ap-7678	37	11	(	(	PUNCT
ap-7678	37	12	lg	lg	PROPN
ap-7678	37	13	-	-	NOUN
ap-7678	37	14	zj	zj	NOUN
ap-7678	37	15	)	)	PUNCT
ap-7678	37	16	in	in	ADP
ap-7678	37	17	the	the	DET
ap-7678	37	18	context	context	NOUN
ap-7678	37	19	of	of	ADP
ap-7678	37	20	their	their	PRON
ap-7678	37	21	order	order	NOUN
ap-7678	37	22	dependent	dependent	ADJ
ap-7678	37	23	conformal	conformal	ADJ
ap-7678	37	24	transformation	transformation	NOUN
ap-7678	37	25	analysis	analysis	NOUN
ap-7678	37	26	[	[	X
ap-7678	37	27	12	12	NUM
ap-7678	37	28	]	]	PUNCT
ap-7678	37	29	.	.	PUNCT
ap-7678	38	1	using	use	VERB
ap-7678	38	2	novel	novel	NOUN
ap-7678	38	3	,	,	PUNCT
ap-7678	38	4	moment	moment	NOUN
ap-7678	38	5	problem	problem	NOUN
ap-7678	38	6	[	[	X
ap-7678	38	7	13	13	NUM
ap-7678	38	8	]	]	PUNCT
ap-7678	38	9	related	relate	VERB
ap-7678	38	10	,	,	PUNCT
ap-7678	38	11	computational	computational	ADJ
ap-7678	38	12	methods	method	NOUN
ap-7678	38	13	,	,	PUNCT
ap-7678	38	14	handy	handy	ADJ
ap-7678	38	15	,	,	PUNCT
ap-7678	38	16	bessis	bessis	NOUN
ap-7678	38	17	,	,	PUNCT
ap-7678	38	18	et	et	PROPN
ap-7678	38	19	al	al	PROPN
ap-7678	38	20	.	.	PUNCT
ap-7678	39	1	[	[	X
ap-7678	39	2	14	14	NUM
ap-7678	39	3	,	,	PUNCT
ap-7678	39	4	15	15	NUM
ap-7678	39	5	]	]	PUNCT
ap-7678	39	6	.	.	PUNCT
ap-7678	40	1	were	be	AUX
ap-7678	40	2	able	able	ADJ
ap-7678	40	3	to	to	PART
ap-7678	40	4	confirm	confirm	VERB
ap-7678	40	5	the	the	DET
ap-7678	40	6	accuracy	accuracy	NOUN
ap-7678	40	7	of	of	ADP
ap-7678	40	8	the	the	DET
ap-7678	40	9	lg	lg	NOUN
ap-7678	40	10	-	-	ADJ
ap-7678	40	11	zj	zj	NOUN
ap-7678	40	12	analysis	analysis	NOUN
ap-7678	40	13	,	,	PUNCT
ap-7678	40	14	by	by	ADP
ap-7678	40	15	computing	compute	VERB
ap-7678	40	16	sufficiently	sufficiently	ADV
ap-7678	40	17	63	63	NUM
ap-7678	40	18	https://doi.org/10.14311/ap.2022.62.0063	https://doi.org/10.14311/ap.2022.62.0063	PROPN
ap-7678	40	19	https://creativecommons.org/licenses/by/4.0/	https://creativecommons.org/licenses/by/4.0/	PROPN
ap-7678	40	20	https://www.cvut.cz/en	https://www.cvut.cz/en	PROPN
ap-7678	40	21	carlos	carlos	PROPN
ap-7678	40	22	r.	r.	PROPN
ap-7678	40	23	handy	handy	PROPN
ap-7678	40	24	acta	acta	PROPN
ap-7678	40	25	polytechnica	polytechnica	PROPN
ap-7678	40	26	tight	tight	ADJ
ap-7678	40	27	bounds	bound	NOUN
ap-7678	40	28	to	to	ADP
ap-7678	40	29	the	the	DET
ap-7678	40	30	ground	ground	NOUN
ap-7678	40	31	state	state	NOUN
ap-7678	40	32	binding	bind	VERB
ap-7678	40	33	energy	energy	NOUN
ap-7678	40	34	.	.	PUNCT
ap-7678	41	1	subsequent	subsequent	ADJ
ap-7678	41	2	studies	study	NOUN
ap-7678	41	3	by	by	ADP
ap-7678	41	4	kravchenko	kravchenko	PROPN
ap-7678	41	5	et	et	PROPN
ap-7678	41	6	al	al	PROPN
ap-7678	41	7	.	.	PUNCT
ap-7678	42	1	[	[	X
ap-7678	42	2	16	16	NUM
ap-7678	42	3	]	]	PUNCT
ap-7678	42	4	,	,	PUNCT
ap-7678	42	5	and	and	CCONJ
ap-7678	42	6	the	the	DET
ap-7678	42	7	more	more	ADV
ap-7678	42	8	recent	recent	ADJ
ap-7678	42	9	work	work	NOUN
ap-7678	42	10	by	by	ADP
ap-7678	42	11	schimerczek	schimerczek	NOUN
ap-7678	42	12	and	and	CCONJ
ap-7678	42	13	wunner	wunner	NOUN
ap-7678	42	14	[	[	X
ap-7678	42	15	17	17	NUM
ap-7678	42	16	]	]	PUNCT
ap-7678	42	17	,	,	PUNCT
ap-7678	42	18	developed	develop	VERB
ap-7678	42	19	a	a	DET
ap-7678	42	20	different	different	ADJ
ap-7678	42	21	formulation	formulation	NOUN
ap-7678	42	22	that	that	PRON
ap-7678	42	23	yielded	yield	VERB
ap-7678	42	24	vastly	vastly	ADV
ap-7678	42	25	improved	improve	VERB
ap-7678	42	26	estimates	estimate	NOUN
ap-7678	42	27	(	(	PUNCT
ap-7678	42	28	not	not	PART
ap-7678	42	29	bounds	bound	NOUN
ap-7678	42	30	)	)	PUNCT
ap-7678	42	31	.	.	PUNCT
ap-7678	43	1	the	the	DET
ap-7678	43	2	work	work	NOUN
ap-7678	43	3	by	by	ADP
ap-7678	43	4	handy	handy	ADJ
ap-7678	43	5	extended	extend	VERB
ap-7678	43	6	oppq	oppq	NOUN
ap-7678	43	7	-	-	PUNCT
ap-7678	43	8	bm	bm	PROPN
ap-7678	43	9	to	to	ADP
ap-7678	43	10	the	the	DET
ap-7678	43	11	qzm	qzm	NOUN
ap-7678	43	12	problem	problem	NOUN
ap-7678	43	13	,	,	PUNCT
ap-7678	43	14	yielding	yield	VERB
ap-7678	43	15	bounds	bound	NOUN
ap-7678	43	16	that	that	PRON
ap-7678	43	17	significantly	significantly	ADV
ap-7678	43	18	improved	improve	VERB
ap-7678	43	19	upon	upon	SCONJ
ap-7678	43	20	,	,	PUNCT
ap-7678	43	21	or	or	CCONJ
ap-7678	43	22	were	be	AUX
ap-7678	43	23	competitive	competitive	ADJ
ap-7678	43	24	with	with	ADP
ap-7678	43	25	,	,	PUNCT
ap-7678	43	26	the	the	DET
ap-7678	43	27	estimates	estimate	NOUN
ap-7678	43	28	by	by	ADP
ap-7678	43	29	kravchenko	kravchenko	PROPN
ap-7678	43	30	et	et	PROPN
ap-7678	43	31	al	al	PROPN
ap-7678	43	32	.	.	PUNCT
ap-7678	44	1	the	the	DET
ap-7678	44	2	qzm	qzm	PROPN
ap-7678	44	3	problem	problem	NOUN
ap-7678	44	4	is	be	AUX
ap-7678	44	5	an	an	DET
ap-7678	44	6	example	example	NOUN
ap-7678	44	7	of	of	ADP
ap-7678	44	8	an	an	DET
ap-7678	44	9	important	important	ADJ
ap-7678	44	10	class	class	NOUN
ap-7678	44	11	of	of	ADP
ap-7678	44	12	problems	problem	NOUN
ap-7678	44	13	for	for	ADP
ap-7678	44	14	which	which	PRON
ap-7678	44	15	bounding	bounding	NOUN
ap-7678	44	16	methods	method	NOUN
ap-7678	44	17	are	be	AUX
ap-7678	44	18	highly	highly	ADV
ap-7678	44	19	relevant	relevant	ADJ
ap-7678	44	20	.	.	PUNCT
ap-7678	45	1	these	these	PRON
ap-7678	45	2	are	be	AUX
ap-7678	45	3	classified	classify	VERB
ap-7678	45	4	as	as	ADP
ap-7678	45	5	singular	singular	ADJ
ap-7678	45	6	perturbation	perturbation	NOUN
ap-7678	45	7	–	–	PUNCT
ap-7678	45	8	strongly	strongly	ADV
ap-7678	45	9	coupled	couple	VERB
ap-7678	45	10	systems	system	NOUN
ap-7678	45	11	.	.	PUNCT
ap-7678	46	1	such	such	ADJ
ap-7678	46	2	systems	system	NOUN
ap-7678	46	3	involve	involve	VERB
ap-7678	46	4	quantum	quantum	ADJ
ap-7678	46	5	particles	particle	NOUN
ap-7678	46	6	subjected	subject	VERB
ap-7678	46	7	to	to	ADP
ap-7678	46	8	very	very	ADV
ap-7678	46	9	strong	strong	ADJ
ap-7678	46	10	forces	force	NOUN
ap-7678	46	11	,	,	PUNCT
ap-7678	46	12	over	over	ADP
ap-7678	46	13	relatively	relatively	ADV
ap-7678	46	14	short	short	ADJ
ap-7678	46	15	length	length	NOUN
ap-7678	46	16	scales	scale	NOUN
ap-7678	46	17	.	.	PUNCT
ap-7678	47	1	this	this	PRON
ap-7678	47	2	is	be	AUX
ap-7678	47	3	inherently	inherently	ADV
ap-7678	47	4	a	a	DET
ap-7678	47	5	multiscale	multiscale	ADJ
ap-7678	47	6	problem	problem	NOUN
ap-7678	47	7	,	,	PUNCT
ap-7678	47	8	in	in	ADP
ap-7678	47	9	keeping	keep	VERB
ap-7678	47	10	with	with	ADP
ap-7678	47	11	the	the	DET
ap-7678	47	12	objectives	objective	NOUN
ap-7678	47	13	of	of	ADP
ap-7678	47	14	wavelet	wavelet	NOUN
ap-7678	47	15	analysis	analysis	NOUN
ap-7678	47	16	[	[	X
ap-7678	47	17	18	18	NUM
ap-7678	47	18	,	,	PUNCT
ap-7678	47	19	19	19	NUM
ap-7678	47	20	]	]	PUNCT
ap-7678	47	21	,	,	PUNCT
ap-7678	47	22	etc	etc	X
ap-7678	47	23	.	.	X
ap-7678	48	1	our	our	PRON
ap-7678	48	2	original	original	ADJ
ap-7678	48	3	immersion	immersion	NOUN
ap-7678	48	4	into	into	ADP
ap-7678	48	5	the	the	DET
ap-7678	48	6	eigenenergy	eigenenergy	NOUN
ap-7678	48	7	bounding	bounding	NOUN
ap-7678	48	8	problem	problem	NOUN
ap-7678	48	9	was	be	AUX
ap-7678	48	10	through	through	ADP
ap-7678	48	11	the	the	DET
ap-7678	48	12	study	study	NOUN
ap-7678	48	13	of	of	ADP
ap-7678	48	14	strong	strong	ADJ
ap-7678	48	15	couplingsingular	couplingsingular	ADJ
ap-7678	48	16	perturbation	perturbation	NOUN
ap-7678	48	17	type	type	NOUN
ap-7678	48	18	systems	system	NOUN
ap-7678	48	19	,	,	PUNCT
ap-7678	48	20	such	such	ADJ
ap-7678	48	21	as	as	ADP
ap-7678	48	22	qzm	qzm	NOUN
ap-7678	48	23	.	.	PUNCT
ap-7678	49	1	the	the	DET
ap-7678	49	2	natural	natural	ADJ
ap-7678	49	3	framework	framework	NOUN
ap-7678	49	4	for	for	ADP
ap-7678	49	5	regulating	regulate	VERB
ap-7678	49	6	these	these	DET
ap-7678	49	7	systems	system	NOUN
ap-7678	49	8	is	be	AUX
ap-7678	49	9	through	through	ADP
ap-7678	49	10	the	the	DET
ap-7678	49	11	use	use	NOUN
ap-7678	49	12	of	of	ADP
ap-7678	49	13	a	a	DET
ap-7678	49	14	non	non	ADJ
ap-7678	49	15	-	-	ADJ
ap-7678	49	16	local	local	ADJ
ap-7678	49	17	,	,	PUNCT
ap-7678	49	18	extensive	extensive	ADJ
ap-7678	49	19	,	,	PUNCT
ap-7678	49	20	representation	representation	NOUN
ap-7678	49	21	.	.	PUNCT
ap-7678	50	1	the	the	DET
ap-7678	50	2	power	power	NOUN
ap-7678	50	3	moments	moment	NOUN
ap-7678	50	4	provide	provide	VERB
ap-7678	50	5	such	such	DET
ap-7678	50	6	a	a	DET
ap-7678	50	7	representation	representation	NOUN
ap-7678	50	8	.	.	PUNCT
ap-7678	51	1	as	as	ADP
ap-7678	51	2	such	such	ADJ
ap-7678	51	3	,	,	PUNCT
ap-7678	51	4	the	the	DET
ap-7678	51	5	types	type	NOUN
ap-7678	51	6	of	of	ADP
ap-7678	51	7	systems	system	NOUN
ap-7678	51	8	studied	study	VERB
ap-7678	51	9	here	here	ADV
ap-7678	51	10	are	be	AUX
ap-7678	51	11	those	those	PRON
ap-7678	51	12	for	for	ADP
ap-7678	51	13	which	which	PRON
ap-7678	51	14	the	the	DET
ap-7678	51	15	schrödinger	schrödinger	ADJ
ap-7678	51	16	operator	operator	NOUN
ap-7678	51	17	configuration	configuration	NOUN
ap-7678	51	18	space	space	NOUN
ap-7678	51	19	problem	problem	NOUN
ap-7678	51	20	can	can	AUX
ap-7678	51	21	be	be	AUX
ap-7678	51	22	transformed	transform	VERB
ap-7678	51	23	into	into	ADP
ap-7678	51	24	a	a	DET
ap-7678	51	25	moments	moment	NOUN
ap-7678	51	26	’	'	PUNCT
ap-7678	51	27	equation	equation	NOUN
ap-7678	51	28	counterpart	counterpart	NOUN
ap-7678	51	29	for	for	ADP
ap-7678	51	30	the	the	DET
ap-7678	51	31	power	power	NOUN
ap-7678	51	32	moments	moment	NOUN
ap-7678	51	33	of	of	ADP
ap-7678	51	34	the	the	DET
ap-7678	51	35	bound	bound	ADJ
ap-7678	51	36	state	state	NOUN
ap-7678	51	37	solutions	solution	NOUN
ap-7678	51	38	.	.	PUNCT
ap-7678	52	1	we	we	PRON
ap-7678	52	2	refer	refer	VERB
ap-7678	52	3	to	to	ADP
ap-7678	52	4	these	these	PRON
ap-7678	52	5	as	as	ADP
ap-7678	52	6	mer	mer	PROPN
ap-7678	52	7	type	type	NOUN
ap-7678	52	8	systems	system	NOUN
ap-7678	52	9	(	(	PUNCT
ap-7678	52	10	i.e.	i.e.	X
ap-7678	52	11	those	those	PRON
ap-7678	52	12	admitting	admit	VERB
ap-7678	52	13	a	a	DET
ap-7678	52	14	moment	moment	NOUN
ap-7678	52	15	equation	equation	NOUN
ap-7678	52	16	representation	representation	NOUN
ap-7678	52	17	)	)	PUNCT
ap-7678	52	18	.	.	PUNCT
ap-7678	53	1	this	this	PRON
ap-7678	53	2	will	will	AUX
ap-7678	53	3	become	become	VERB
ap-7678	53	4	clear	clear	ADV
ap-7678	53	5	below	below	ADV
ap-7678	53	6	.	.	PUNCT
ap-7678	54	1	this	this	DET
ap-7678	54	2	condition	condition	NOUN
ap-7678	54	3	can	can	AUX
ap-7678	54	4	be	be	AUX
ap-7678	54	5	relaxed	relax	VERB
ap-7678	54	6	,	,	PUNCT
ap-7678	54	7	and	and	CCONJ
ap-7678	54	8	essentially	essentially	ADV
ap-7678	54	9	imposed	impose	VERB
ap-7678	54	10	on	on	ADP
ap-7678	54	11	systems	system	NOUN
ap-7678	54	12	that	that	PRON
ap-7678	54	13	do	do	AUX
ap-7678	54	14	not	not	PART
ap-7678	54	15	admit	admit	VERB
ap-7678	54	16	such	such	ADJ
ap-7678	54	17	mer	mer	NOUN
ap-7678	54	18	formulations	formulation	NOUN
ap-7678	54	19	,	,	PUNCT
ap-7678	54	20	making	make	VERB
ap-7678	54	21	the	the	DET
ap-7678	54	22	underlying	underlie	VERB
ap-7678	54	23	oppq	oppq	PROPN
ap-7678	54	24	-	-	PUNCT
ap-7678	54	25	bm	bm	PROPN
ap-7678	54	26	principles	principle	NOUN
ap-7678	54	27	applicable	applicable	ADJ
ap-7678	54	28	to	to	ADP
ap-7678	54	29	many	many	ADJ
ap-7678	54	30	different	different	ADJ
ap-7678	54	31	types	type	NOUN
ap-7678	54	32	of	of	ADP
ap-7678	54	33	systems	system	NOUN
ap-7678	54	34	.	.	PUNCT
ap-7678	55	1	the	the	DET
ap-7678	55	2	details	detail	NOUN
ap-7678	55	3	of	of	ADP
ap-7678	55	4	this	this	DET
ap-7678	55	5	expanded	expand	VERB
ap-7678	55	6	analysis	analysis	NOUN
ap-7678	55	7	will	will	AUX
ap-7678	55	8	be	be	AUX
ap-7678	55	9	communicated	communicate	VERB
ap-7678	55	10	shortly	shortly	ADV
ap-7678	55	11	.	.	PUNCT
ap-7678	56	1	1.1	1.1	NUM
ap-7678	56	2	.	.	PUNCT
ap-7678	56	3	singular	singular	NOUN
ap-7678	56	4	-	-	PUNCT
ap-7678	56	5	perturbation	perturbation	NOUN
ap-7678	56	6	strong	strong	ADJ
ap-7678	56	7	coupling	coupling	NOUN
ap-7678	56	8	problems	problem	NOUN
ap-7678	56	9	and	and	CCONJ
ap-7678	56	10	moment	moment	NOUN
ap-7678	56	11	representations	representation	NOUN
ap-7678	56	12	it	it	PRON
ap-7678	56	13	is	be	AUX
ap-7678	56	14	not	not	PART
ap-7678	56	15	widely	widely	ADV
ap-7678	56	16	appreciated	appreciated	ADJ
ap-7678	56	17	that	that	SCONJ
ap-7678	56	18	power	power	NOUN
ap-7678	56	19	moments	moment	NOUN
ap-7678	56	20	are	be	AUX
ap-7678	56	21	relevant	relevant	ADJ
ap-7678	56	22	in	in	ADP
ap-7678	56	23	understanding	understand	VERB
ap-7678	56	24	the	the	DET
ap-7678	56	25	multiscale	multiscale	ADJ
ap-7678	56	26	structure	structure	NOUN
ap-7678	56	27	of	of	ADP
ap-7678	56	28	most	most	ADJ
ap-7678	56	29	systems	system	NOUN
ap-7678	56	30	.	.	PUNCT
ap-7678	57	1	thus	thus	ADV
ap-7678	57	2	,	,	PUNCT
ap-7678	57	3	consider	consider	VERB
ap-7678	57	4	the	the	DET
ap-7678	57	5	scaling	scaling	NOUN
ap-7678	57	6	transform	transform	NOUN
ap-7678	57	7	of	of	ADP
ap-7678	57	8	a	a	DET
ap-7678	57	9	given	give	VERB
ap-7678	57	10	wavefunction	wavefunction	NOUN
ap-7678	57	11	(	(	PUNCT
ap-7678	57	12	i.e.	i.e.	X
ap-7678	57	13	signal	signal	NOUN
ap-7678	57	14	,	,	PUNCT
ap-7678	57	15	where	where	SCONJ
ap-7678	57	16	we	we	PRON
ap-7678	57	17	have	have	AUX
ap-7678	57	18	simplified	simplify	VERB
ap-7678	57	19	the	the	DET
ap-7678	57	20	notation	notation	NOUN
ap-7678	57	21	to	to	ADP
ap-7678	57	22	that	that	PRON
ap-7678	57	23	of	of	ADP
ap-7678	57	24	one	one	NUM
ap-7678	57	25	dimension	dimension	NOUN
ap-7678	57	26	):	):	PUNCT
ap-7678	57	27	sψ(a	sψ(a	NOUN
ap-7678	57	28	,	,	PUNCT
ap-7678	57	29	b	b	NOUN
ap-7678	57	30	)	)	PUNCT
ap-7678	57	31	≡	≡	PROPN
ap-7678	57	32	1	1	NUM
ap-7678	58	1	aν	aν	NOUN
ap-7678	58	2	∫	∫	PROPN
ap-7678	59	1	+	+	CCONJ
ap-7678	59	2	∞	∞	PROPN
ap-7678	59	3	−∞	−∞	ADP
ap-7678	59	4	dx	dx	PROPN
ap-7678	59	5	s(x	s(x	PROPN
ap-7678	59	6	−	−	PROPN
ap-7678	59	7	b	b	PROPN
ap-7678	59	8	a	a	PRON
ap-7678	59	9	)	)	PUNCT
ap-7678	59	10	ψ(x	ψ(x	NOUN
ap-7678	59	11	)	)	PUNCT
ap-7678	59	12	,	,	PUNCT
ap-7678	59	13	(	(	PUNCT
ap-7678	59	14	2	2	X
ap-7678	59	15	)	)	PUNCT
ap-7678	59	16	where	where	SCONJ
ap-7678	59	17	lima→0+sψ(a	lima→0+sψ(a	PROPN
ap-7678	59	18	,	,	PUNCT
ap-7678	59	19	b	b	NOUN
ap-7678	59	20	)	)	PUNCT
ap-7678	59	21	=	=	SYM
ap-7678	59	22	ψ(b	ψ(b	NOUN
ap-7678	59	23	)	)	PUNCT
ap-7678	59	24	,	,	PUNCT
ap-7678	59	25	(	(	PUNCT
ap-7678	59	26	3	3	X
ap-7678	59	27	)	)	PUNCT
ap-7678	59	28	and	and	CCONJ
ap-7678	59	29	ν	ν	PROPN
ap-7678	59	30	≡	≡	PROPN
ap-7678	59	31	∫	∫	PROPN
ap-7678	59	32	dx	dx	PROPN
ap-7678	59	33	s(x	s(x	PROPN
ap-7678	59	34	)	)	PUNCT
ap-7678	59	35	̸=	̸=	PROPN
ap-7678	59	36	0	0	NUM
ap-7678	59	37	.	.	PUNCT
ap-7678	60	1	physicists	physicist	NOUN
ap-7678	60	2	are	be	AUX
ap-7678	60	3	biased	bias	VERB
ap-7678	60	4	in	in	ADP
ap-7678	60	5	favor	favor	NOUN
ap-7678	60	6	of	of	ADP
ap-7678	60	7	attaining	attain	VERB
ap-7678	60	8	an	an	DET
ap-7678	60	9	analytical	analytical	ADJ
ap-7678	60	10	understanding	understanding	NOUN
ap-7678	60	11	of	of	ADP
ap-7678	60	12	problems	problem	NOUN
ap-7678	60	13	.	.	PUNCT
ap-7678	61	1	therefore	therefore	ADV
ap-7678	61	2	the	the	DET
ap-7678	61	3	natural	natural	ADJ
ap-7678	61	4	question	question	NOUN
ap-7678	61	5	to	to	PART
ap-7678	61	6	ask	ask	VERB
ap-7678	61	7	is	be	AUX
ap-7678	61	8	,	,	PUNCT
ap-7678	61	9	what	what	PRON
ap-7678	61	10	is	be	AUX
ap-7678	61	11	the	the	DET
ap-7678	61	12	analytical	analytical	ADJ
ap-7678	61	13	dependence	dependence	NOUN
ap-7678	61	14	in	in	ADP
ap-7678	61	15	the	the	DET
ap-7678	61	16	inverse	inverse	NOUN
ap-7678	61	17	scale	scale	NOUN
ap-7678	61	18	(	(	PUNCT
ap-7678	61	19	i.e.	i.e.	X
ap-7678	61	20	1	1	NUM
ap-7678	61	21	a	a	NOUN
ap-7678	61	22	)	)	PUNCT
ap-7678	61	23	for	for	ADP
ap-7678	61	24	this	this	DET
ap-7678	61	25	scaling	scaling	NOUN
ap-7678	61	26	transform	transform	NOUN
ap-7678	61	27	,	,	PUNCT
ap-7678	61	28	if	if	SCONJ
ap-7678	61	29	ψ	ψ	NOUN
ap-7678	61	30	is	be	AUX
ap-7678	61	31	a	a	DET
ap-7678	61	32	bounded	bound	VERB
ap-7678	61	33	,	,	PUNCT
ap-7678	61	34	l2	l2	NOUN
ap-7678	61	35	state	state	NOUN
ap-7678	61	36	?	?	PUNCT
ap-7678	62	1	it	it	PRON
ap-7678	62	2	will	will	AUX
ap-7678	62	3	become	become	VERB
ap-7678	62	4	clear	clear	ADJ
ap-7678	62	5	from	from	ADP
ap-7678	62	6	eq	eq	ADP
ap-7678	62	7	.	.	PUNCT
ap-7678	63	1	(	(	PUNCT
ap-7678	63	2	4	4	NUM
ap-7678	63	3	)	)	PUNCT
ap-7678	63	4	,	,	PUNCT
ap-7678	63	5	that	that	SCONJ
ap-7678	63	6	the	the	DET
ap-7678	63	7	power	power	NOUN
ap-7678	63	8	moments	moment	NOUN
ap-7678	63	9	of	of	ADP
ap-7678	63	10	the	the	DET
ap-7678	63	11	bound	bound	ADJ
ap-7678	63	12	state	state	NOUN
ap-7678	63	13	solution	solution	NOUN
ap-7678	63	14	,	,	PUNCT
ap-7678	63	15	determine	determine	VERB
ap-7678	63	16	the	the	DET
ap-7678	63	17	analytic	analytic	ADJ
ap-7678	63	18	structure	structure	NOUN
ap-7678	63	19	of	of	ADP
ap-7678	63	20	the	the	DET
ap-7678	63	21	inverse	inverse	ADJ
ap-7678	63	22	scale	scale	NOUN
ap-7678	63	23	expansion	expansion	NOUN
ap-7678	63	24	.	.	PUNCT
ap-7678	64	1	these	these	DET
ap-7678	64	2	considerations	consideration	NOUN
ap-7678	64	3	underlie	underlie	VERB
ap-7678	64	4	the	the	DET
ap-7678	64	5	analysis	analysis	NOUN
ap-7678	64	6	by	by	ADP
ap-7678	64	7	handy	handy	ADJ
ap-7678	64	8	[	[	X
ap-7678	64	9	20	20	NUM
ap-7678	64	10	]	]	PUNCT
ap-7678	64	11	.	.	PUNCT
ap-7678	65	1	alternatively	alternatively	ADV
ap-7678	65	2	,	,	PUNCT
ap-7678	65	3	engineers	engineer	NOUN
ap-7678	65	4	are	be	AUX
ap-7678	65	5	more	more	ADV
ap-7678	65	6	oriented	oriented	ADJ
ap-7678	65	7	towards	towards	ADP
ap-7678	65	8	computational	computational	ADJ
ap-7678	65	9	capabilities	capability	NOUN
ap-7678	65	10	.	.	PUNCT
ap-7678	66	1	if	if	SCONJ
ap-7678	66	2	so	so	ADV
ap-7678	66	3	,	,	PUNCT
ap-7678	66	4	then	then	ADV
ap-7678	66	5	it	it	PRON
ap-7678	66	6	readily	readily	ADV
ap-7678	66	7	follows	follow	VERB
ap-7678	66	8	that	that	SCONJ
ap-7678	66	9	the	the	DET
ap-7678	66	10	a	a	DET
ap-7678	66	11	→	→	SYM
ap-7678	66	12	0	0	NUM
ap-7678	66	13	limit	limit	NOUN
ap-7678	66	14	can	can	AUX
ap-7678	66	15	be	be	AUX
ap-7678	66	16	replaced	replace	VERB
ap-7678	66	17	by	by	ADP
ap-7678	66	18	the	the	DET
ap-7678	66	19	integral	integral	ADJ
ap-7678	66	20	∫	∫	PROPN
ap-7678	66	21	∞	∞	PROPN
ap-7678	66	22	0	0	PUNCT
ap-7678	67	1	dα∂α	dα∂α	INTJ
ap-7678	67	2	(	(	PUNCT
ap-7678	67	3	αs	αs	INTJ
ap-7678	67	4	(	(	PUNCT
ap-7678	67	5	α(x	α(x	PROPN
ap-7678	67	6	−	−	PROPN
ap-7678	67	7	b	b	NOUN
ap-7678	67	8	)	)	PUNCT
ap-7678	67	9	)	)	PUNCT
ap-7678	67	10	)	)	PUNCT
ap-7678	67	11	,	,	PUNCT
ap-7678	67	12	where	where	SCONJ
ap-7678	67	13	α	α	PROPN
ap-7678	67	14	≡	≡	PROPN
ap-7678	67	15	1	1	NUM
ap-7678	67	16	a	a	PRON
ap-7678	67	17	,	,	PUNCT
ap-7678	67	18	which	which	PRON
ap-7678	67	19	after	after	ADP
ap-7678	67	20	a	a	DET
ap-7678	67	21	convolution	convolution	NOUN
ap-7678	67	22	substitution	substitution	NOUN
ap-7678	67	23	gives	give	VERB
ap-7678	67	24	one	one	NUM
ap-7678	67	25	the	the	DET
ap-7678	67	26	continuous	continuous	ADJ
ap-7678	67	27	wavelet	wavelet	NOUN
ap-7678	67	28	transform	transform	NOUN
ap-7678	67	29	[	[	X
ap-7678	67	30	19	19	NUM
ap-7678	67	31	]	]	PUNCT
ap-7678	67	32	.	.	PUNCT
ap-7678	68	1	for	for	ADP
ap-7678	68	2	bound	bind	VERB
ap-7678	68	3	state	state	NOUN
ap-7678	68	4	configurations	configuration	NOUN
ap-7678	68	5	,	,	PUNCT
ap-7678	68	6	if	if	SCONJ
ap-7678	68	7	lim|x|→∞	lim|x|→∞	NOUN
ap-7678	68	8	ψ(|x|)s(|x|eiθ	ψ(|x|)s(|x|eiθ	PROPN
ap-7678	68	9	)	)	PUNCT
ap-7678	68	10	=	=	SYM
ap-7678	68	11	0	0	NUM
ap-7678	68	12	,	,	PUNCT
ap-7678	68	13	exponentially	exponentially	ADV
ap-7678	68	14	,	,	PUNCT
ap-7678	68	15	for	for	ADP
ap-7678	68	16	arbitrary	arbitrary	ADJ
ap-7678	68	17	θ	θ	NOUN
ap-7678	68	18	,	,	PUNCT
ap-7678	68	19	then	then	ADV
ap-7678	68	20	the	the	DET
ap-7678	68	21	scaling	scaling	NOUN
ap-7678	68	22	transform	transform	NOUN
ap-7678	68	23	becomes	become	VERB
ap-7678	68	24	analytic	analytic	ADJ
ap-7678	68	25	in	in	ADP
ap-7678	68	26	the	the	DET
ap-7678	68	27	inverse	inverse	NOUN
ap-7678	68	28	scale	scale	NOUN
ap-7678	68	29	,	,	PUNCT
ap-7678	68	30	1	1	NUM
ap-7678	68	31	a	a	PRON
ap-7678	68	32	.	.	PUNCT
ap-7678	69	1	under	under	ADP
ap-7678	69	2	this	this	DET
ap-7678	69	3	assumption	assumption	NOUN
ap-7678	69	4	,	,	PUNCT
ap-7678	69	5	the	the	DET
ap-7678	69	6	scaling	scaling	NOUN
ap-7678	69	7	transform	transform	VERB
ap-7678	69	8	’s	’s	PART
ap-7678	69	9	analytic	analytic	ADJ
ap-7678	69	10	expansion	expansion	NOUN
ap-7678	69	11	depends	depend	VERB
ap-7678	69	12	on	on	ADP
ap-7678	69	13	the	the	DET
ap-7678	69	14	moments	moment	NOUN
ap-7678	70	1	[	[	X
ap-7678	70	2	20	20	NUM
ap-7678	70	3	]	]	PUNCT
ap-7678	70	4	sψ(a	sψ(a	NOUN
ap-7678	70	5	,	,	PUNCT
ap-7678	70	6	b	b	NOUN
ap-7678	70	7	)	)	PUNCT
ap-7678	70	8	=	=	SYM
ap-7678	70	9	1	1	NUM
ap-7678	70	10	aν	aν	NOUN
ap-7678	70	11	∞∑	∞∑	NUM
ap-7678	70	12	j=0	j=0	PROPN
ap-7678	70	13	σj	σj	VERB
ap-7678	70	14	j!aj	j!aj	PROPN
ap-7678	70	15	j∑	j∑	PROPN
ap-7678	70	16	p=0	p=0	PROPN
ap-7678	71	1	(	(	PUNCT
ap-7678	71	2	j	j	PROPN
ap-7678	71	3	p	p	NOUN
ap-7678	71	4	)	)	PUNCT
ap-7678	71	5	(	(	PUNCT
ap-7678	71	6	−b)j−pµ(p	−b)j−pµ(p	X
ap-7678	71	7	)	)	PUNCT
ap-7678	71	8	,	,	PUNCT
ap-7678	71	9	(	(	PUNCT
ap-7678	71	10	4	4	X
ap-7678	71	11	)	)	PUNCT
ap-7678	71	12	where	where	SCONJ
ap-7678	71	13	∂j	∂j	PROPN
ap-7678	71	14	xs(0	xs(0	PROPN
ap-7678	71	15	)	)	PUNCT
ap-7678	71	16	≡	≡	PROPN
ap-7678	71	17	σj	σj	VERB
ap-7678	71	18	,	,	PUNCT
ap-7678	71	19	and	and	CCONJ
ap-7678	71	20	µ(p	µ(p	ADJ
ap-7678	71	21	)	)	PUNCT
ap-7678	72	1	≡	≡	PROPN
ap-7678	72	2	∫	∫	PROPN
ap-7678	73	1	+	+	NUM
ap-7678	73	2	∞	∞	PROPN
ap-7678	73	3	−∞	−∞	ADP
ap-7678	73	4	dx	dx	PROPN
ap-7678	73	5	xpψ(x	xpψ(x	PROPN
ap-7678	73	6	)	)	PUNCT
ap-7678	73	7	.	.	PUNCT
ap-7678	74	1	(	(	PUNCT
ap-7678	74	2	5	5	X
ap-7678	74	3	)	)	PUNCT
ap-7678	74	4	2	2	NUM
ap-7678	74	5	.	.	PUNCT
ap-7678	75	1	the	the	DET
ap-7678	75	2	moment	moment	NOUN
ap-7678	75	3	equation	equation	NOUN
ap-7678	75	4	representation	representation	NOUN
ap-7678	75	5	the	the	DET
ap-7678	75	6	natural	natural	ADJ
ap-7678	75	7	extension	extension	NOUN
ap-7678	75	8	of	of	ADP
ap-7678	75	9	the	the	DET
ap-7678	75	10	above	above	ADJ
ap-7678	75	11	considerations	consideration	NOUN
ap-7678	75	12	is	be	AUX
ap-7678	75	13	to	to	PART
ap-7678	75	14	study	study	VERB
ap-7678	75	15	linear	linear	ADJ
ap-7678	75	16	quantum	quantum	NOUN
ap-7678	75	17	systems	system	NOUN
ap-7678	75	18	whose	whose	DET
ap-7678	75	19	differential	differential	ADJ
ap-7678	75	20	form	form	NOUN
ap-7678	75	21	is	be	AUX
ap-7678	75	22	transformable	transformable	ADJ
ap-7678	75	23	into	into	ADP
ap-7678	75	24	a	a	DET
ap-7678	75	25	moment	moment	NOUN
ap-7678	75	26	equation	equation	NOUN
ap-7678	75	27	.	.	PUNCT
ap-7678	76	1	thus	thus	ADV
ap-7678	76	2	consider	consider	VERB
ap-7678	76	3	the	the	DET
ap-7678	76	4	sextic	sextic	ADJ
ap-7678	76	5	anharmonic	anharmonic	ADJ
ap-7678	76	6	oscillator	oscillator	NOUN
ap-7678	76	7	potential	potential	NOUN
ap-7678	76	8	,	,	PUNCT
ap-7678	76	9	where	where	SCONJ
ap-7678	76	10	the	the	DET
ap-7678	76	11	physical	physical	ADJ
ap-7678	76	12	parameters	parameter	NOUN
ap-7678	76	13	have	have	AUX
ap-7678	76	14	been	be	AUX
ap-7678	76	15	re	re	VERB
ap-7678	76	16	-	-	VERB
ap-7678	76	17	scaled	scale	VERB
ap-7678	76	18	:	:	PUNCT
ap-7678	76	19	−ϵ2∂2	−ϵ2∂2	ADP
ap-7678	76	20	xψ(x	xψ(x	NUM
ap-7678	76	21	)	)	PUNCT
ap-7678	77	1	+	+	CCONJ
ap-7678	77	2	(	(	PUNCT
ap-7678	77	3	mx2	mx2	NOUN
ap-7678	77	4	+	+	CCONJ
ap-7678	77	5	gx6)ψ(x	gx6)ψ(x	NOUN
ap-7678	77	6	)	)	PUNCT
ap-7678	78	1	=	=	SYM
ap-7678	78	2	eψ(x	eψ(x	NOUN
ap-7678	78	3	)	)	PUNCT
ap-7678	78	4	.	.	PUNCT
ap-7678	79	1	(	(	PUNCT
ap-7678	79	2	6	6	X
ap-7678	79	3	)	)	PUNCT
ap-7678	79	4	the	the	DET
ap-7678	79	5	nature	nature	NOUN
ap-7678	79	6	of	of	ADP
ap-7678	79	7	physical	physical	ADJ
ap-7678	79	8	quantum	quantum	NOUN
ap-7678	79	9	systems	system	NOUN
ap-7678	79	10	is	be	AUX
ap-7678	79	11	that	that	SCONJ
ap-7678	79	12	the	the	DET
ap-7678	79	13	discrete	discrete	ADJ
ap-7678	79	14	states	state	NOUN
ap-7678	79	15	decay	decay	VERB
ap-7678	79	16	exponentially	exponentially	ADV
ap-7678	79	17	,	,	PUNCT
ap-7678	79	18	and	and	CCONJ
ap-7678	79	19	therefore	therefore	ADV
ap-7678	79	20	have	have	VERB
ap-7678	79	21	finite	finite	ADJ
ap-7678	79	22	power	power	NOUN
ap-7678	79	23	moments	moment	NOUN
ap-7678	79	24	.	.	PUNCT
ap-7678	80	1	the	the	DET
ap-7678	80	2	unphysical	unphysical	ADJ
ap-7678	80	3	solutions	solution	NOUN
ap-7678	80	4	become	become	VERB
ap-7678	80	5	exponentially	exponentially	ADV
ap-7678	80	6	unbounded	unbounded	ADJ
ap-7678	80	7	in	in	ADP
ap-7678	80	8	one	one	NUM
ap-7678	80	9	or	or	CCONJ
ap-7678	80	10	both	both	CCONJ
ap-7678	80	11	asymptotic	asymptotic	ADJ
ap-7678	80	12	directions	direction	NOUN
ap-7678	80	13	,	,	PUNCT
ap-7678	80	14	therefore	therefore	ADV
ap-7678	80	15	their	their	PRON
ap-7678	80	16	power	power	NOUN
ap-7678	80	17	moments	moment	NOUN
ap-7678	80	18	are	be	AUX
ap-7678	80	19	infinite	infinite	ADJ
ap-7678	80	20	.	.	PUNCT
ap-7678	81	1	we	we	PRON
ap-7678	81	2	can	can	AUX
ap-7678	81	3	multiply	multiply	VERB
ap-7678	81	4	the	the	DET
ap-7678	81	5	above	above	ADJ
ap-7678	81	6	equation	equation	NOUN
ap-7678	81	7	by	by	ADP
ap-7678	81	8	xp	xp	ADV
ap-7678	81	9	and	and	CCONJ
ap-7678	81	10	integrate	integrate	VERB
ap-7678	81	11	by	by	ADP
ap-7678	81	12	parts	part	NOUN
ap-7678	81	13	,	,	PUNCT
ap-7678	81	14	assuming	assume	VERB
ap-7678	81	15	the	the	DET
ap-7678	81	16	underlying	underlie	VERB
ap-7678	81	17	wavefunction	wavefunction	NOUN
ap-7678	81	18	is	be	AUX
ap-7678	81	19	that	that	PRON
ap-7678	81	20	of	of	ADP
ap-7678	81	21	a	a	DET
ap-7678	81	22	discrete	discrete	ADJ
ap-7678	81	23	state	state	NOUN
ap-7678	81	24	.	.	PUNCT
ap-7678	82	1	we	we	PRON
ap-7678	82	2	then	then	ADV
ap-7678	82	3	obtain	obtain	VERB
ap-7678	82	4	the	the	DET
ap-7678	82	5	moment	moment	NOUN
ap-7678	82	6	equation	equation	NOUN
ap-7678	82	7	representation	representation	NOUN
ap-7678	82	8	(	(	PUNCT
ap-7678	82	9	mer	mer	ADJ
ap-7678	82	10	):	):	PUNCT
ap-7678	82	11	gµ(p	gµ(p	NOUN
ap-7678	82	12	+	+	NOUN
ap-7678	82	13	6	6	NUM
ap-7678	82	14	)	)	PUNCT
ap-7678	82	15	=	=	SYM
ap-7678	82	16	−	−	NOUN
ap-7678	82	17	mµ(p	mµ(p	NOUN
ap-7678	83	1	+	+	CCONJ
ap-7678	83	2	2	2	X
ap-7678	83	3	)	)	PUNCT
ap-7678	83	4	+	+	CCONJ
ap-7678	83	5	eµ(p	eµ(p	NUM
ap-7678	83	6	)	)	PUNCT
ap-7678	84	1	+	+	CCONJ
ap-7678	84	2	p(p	p(p	ADV
ap-7678	84	3	−	−	NOUN
ap-7678	84	4	1)ϵ2µ(p	1)ϵ2µ(p	NUM
ap-7678	84	5	−	−	NOUN
ap-7678	84	6	2	2	NUM
ap-7678	84	7	)	)	PUNCT
ap-7678	84	8	,	,	PUNCT
ap-7678	84	9	p	p	PRON
ap-7678	84	10	≥	≥	NOUN
ap-7678	84	11	0	0	NUM
ap-7678	84	12	.	.	PUNCT
ap-7678	85	1	(	(	PUNCT
ap-7678	85	2	7	7	X
ap-7678	85	3	)	)	PUNCT
ap-7678	85	4	this	this	DET
ap-7678	85	5	homogeneous	homogeneous	ADJ
ap-7678	85	6	mer	mer	NOUN
ap-7678	85	7	expression	expression	NOUN
ap-7678	85	8	is	be	AUX
ap-7678	85	9	a	a	DET
ap-7678	85	10	finite	finite	ADJ
ap-7678	85	11	difference	difference	NOUN
ap-7678	85	12	equation	equation	NOUN
ap-7678	85	13	of	of	ADP
ap-7678	85	14	effective	effective	ADJ
ap-7678	85	15	order	order	NOUN
ap-7678	85	16	1	1	NUM
ap-7678	85	17	+	+	CCONJ
ap-7678	85	18	ms	ms	NOUN
ap-7678	85	19	where	where	SCONJ
ap-7678	85	20	ms	ms	NOUN
ap-7678	85	21	=	=	PROPN
ap-7678	85	22	5	5	PROPN
ap-7678	85	23	.	.	PUNCT
ap-7678	85	24	that	that	PRON
ap-7678	85	25	is	be	AUX
ap-7678	85	26	,	,	PUNCT
ap-7678	85	27	for	for	ADP
ap-7678	85	28	any	any	DET
ap-7678	85	29	e	e	NOUN
ap-7678	85	30	parameter	parameter	NOUN
ap-7678	85	31	value	value	NOUN
ap-7678	85	32	,	,	PUNCT
ap-7678	85	33	the	the	DET
ap-7678	85	34	first	first	ADJ
ap-7678	85	35	six	six	NUM
ap-7678	85	36	power	power	NOUN
ap-7678	85	37	moments	moment	NOUN
ap-7678	85	38	{	{	PUNCT
ap-7678	85	39	µ0	µ0	PROPN
ap-7678	85	40	,	,	PUNCT
ap-7678	85	41	µ1	µ1	PROPN
ap-7678	85	42	,	,	PUNCT
ap-7678	85	43	.	.	PUNCT
ap-7678	85	44	.	.	PUNCT
ap-7678	86	1	.	.	PUNCT
ap-7678	87	1	,	,	PUNCT
ap-7678	87	2	µ5	µ5	PROPN
ap-7678	87	3	}	}	PUNCT
ap-7678	87	4	(	(	PUNCT
ap-7678	87	5	i.e.	i.e.	X
ap-7678	87	6	µ(ℓ	µ(ℓ	NOUN
ap-7678	87	7	)	)	PUNCT
ap-7678	87	8	≡	≡	PROPN
ap-7678	87	9	µℓ	µℓ	PART
ap-7678	87	10	)	)	PUNCT
ap-7678	87	11	are	be	AUX
ap-7678	87	12	the	the	DET
ap-7678	87	13	initialization	initialization	NOUN
ap-7678	87	14	moments	moment	NOUN
ap-7678	87	15	,	,	PUNCT
ap-7678	87	16	or	or	CCONJ
ap-7678	87	17	missing	miss	VERB
ap-7678	87	18	moments	moment	NOUN
ap-7678	87	19	,	,	PUNCT
ap-7678	87	20	and	and	CCONJ
ap-7678	87	21	generate	generate	VERB
ap-7678	87	22	all	all	DET
ap-7678	87	23	the	the	DET
ap-7678	87	24	other	other	ADJ
ap-7678	87	25	power	power	NOUN
ap-7678	87	26	moments	moment	NOUN
ap-7678	87	27	through	through	ADP
ap-7678	87	28	closed	close	VERB
ap-7678	87	29	form	form	NOUN
ap-7678	87	30	,	,	PUNCT
ap-7678	87	31	energy	energy	NOUN
ap-7678	87	32	dependent	dependent	ADJ
ap-7678	87	33	coefficients	coefficient	NOUN
ap-7678	87	34	:	:	PUNCT
ap-7678	87	35	µ(p	µ(p	ADJ
ap-7678	87	36	)	)	PUNCT
ap-7678	87	37	=	=	SYM
ap-7678	88	1	ms∑	ms∑	PROPN
ap-7678	88	2	ℓ=0	ℓ=0	SYM
ap-7678	88	3	me(p	me(p	NOUN
ap-7678	88	4	,	,	PUNCT
ap-7678	88	5	ℓ	ℓ	NOUN
ap-7678	88	6	)	)	PUNCT
ap-7678	88	7	µℓ	µℓ	ADP
ap-7678	88	8	,	,	PUNCT
ap-7678	88	9	p	p	PRON
ap-7678	88	10	≥	≥	NOUN
ap-7678	88	11	0	0	NUM
ap-7678	88	12	.	.	PUNCT
ap-7678	89	1	(	(	PUNCT
ap-7678	89	2	8)	8)	NUM
ap-7678	89	3	64	64	NUM
ap-7678	89	4	vol	vol	NOUN
ap-7678	89	5	.	.	PUNCT
ap-7678	90	1	62	62	NUM
ap-7678	90	2	no	no	INTJ
ap-7678	90	3	.	.	PUNCT
ap-7678	91	1	1/2022	1/2022	NUM
ap-7678	91	2	algebraic	algebraic	ADJ
ap-7678	91	3	eigenenergy	eigenenergy	NOUN
ap-7678	91	4	bounding	bounding	NOUN
ap-7678	91	5	method	method	NOUN
ap-7678	91	6	if	if	SCONJ
ap-7678	91	7	the	the	DET
ap-7678	91	8	coupling	coupling	NOUN
ap-7678	91	9	strength	strength	NOUN
ap-7678	91	10	is	be	AUX
ap-7678	91	11	large	large	ADJ
ap-7678	91	12	,	,	PUNCT
ap-7678	91	13	g	g	PROPN
ap-7678	91	14	>	>	X
ap-7678	91	15	>	>	X
ap-7678	91	16	1	1	NUM
ap-7678	91	17	,	,	PUNCT
ap-7678	91	18	the	the	DET
ap-7678	91	19	natural	natural	ADJ
ap-7678	91	20	inclination	inclination	NOUN
ap-7678	91	21	is	be	AUX
ap-7678	91	22	to	to	PART
ap-7678	91	23	attempt	attempt	VERB
ap-7678	91	24	some	some	DET
ap-7678	91	25	kind	kind	NOUN
ap-7678	91	26	of	of	ADP
ap-7678	91	27	singular	singular	ADJ
ap-7678	91	28	perturbation	perturbation	NOUN
ap-7678	91	29	analysis	analysis	NOUN
ap-7678	91	30	involving	involve	VERB
ap-7678	91	31	expansions	expansion	NOUN
ap-7678	91	32	around	around	ADP
ap-7678	91	33	the	the	DET
ap-7678	91	34	kinetic	kinetic	ADJ
ap-7678	91	35	energy	energy	NOUN
ap-7678	91	36	term	term	NOUN
ap-7678	91	37	;	;	PUNCT
ap-7678	91	38	or	or	CCONJ
ap-7678	91	39	,	,	PUNCT
ap-7678	91	40	alternatively	alternatively	ADV
ap-7678	91	41	,	,	PUNCT
ap-7678	91	42	a	a	DET
ap-7678	91	43	large	large	ADJ
ap-7678	91	44	perturbative	perturbative	ADJ
ap-7678	91	45	expansion	expansion	NOUN
ap-7678	91	46	/	/	SYM
ap-7678	91	47	resummation	resummation	NOUN
ap-7678	91	48	analysis	analysis	NOUN
ap-7678	91	49	.	.	PUNCT
ap-7678	92	1	in	in	ADP
ap-7678	92	2	configurations	configuration	NOUN
ap-7678	92	3	space	space	NOUN
ap-7678	92	4	,	,	PUNCT
ap-7678	92	5	kinetic	kinetic	ADJ
ap-7678	92	6	energy	energy	NOUN
ap-7678	92	7	expansions	expansion	NOUN
ap-7678	92	8	become	become	VERB
ap-7678	92	9	singular	singular	ADJ
ap-7678	92	10	(	(	PUNCT
ap-7678	92	11	i.e.	i.e.	X
ap-7678	92	12	expanding	expand	VERB
ap-7678	92	13	in	in	ADP
ap-7678	92	14	ϵ2	ϵ2	PROPN
ap-7678	92	15	)	)	PUNCT
ap-7678	92	16	because	because	SCONJ
ap-7678	92	17	the	the	DET
ap-7678	92	18	order	order	NOUN
ap-7678	92	19	of	of	ADP
ap-7678	92	20	the	the	DET
ap-7678	92	21	differential	differential	ADJ
ap-7678	92	22	equation	equation	NOUN
ap-7678	92	23	abruptly	abruptly	ADV
ap-7678	92	24	changes	change	VERB
ap-7678	92	25	from	from	ADP
ap-7678	92	26	zero	zero	NUM
ap-7678	92	27	to	to	ADP
ap-7678	92	28	two	two	NUM
ap-7678	92	29	.	.	PUNCT
ap-7678	93	1	however	however	ADV
ap-7678	93	2	,	,	PUNCT
ap-7678	93	3	the	the	DET
ap-7678	93	4	order	order	NOUN
ap-7678	93	5	of	of	ADP
ap-7678	93	6	the	the	DET
ap-7678	93	7	mer	mer	PROPN
ap-7678	93	8	relation	relation	NOUN
ap-7678	93	9	does	do	AUX
ap-7678	93	10	not	not	PART
ap-7678	93	11	change	change	VERB
ap-7678	93	12	as	as	SCONJ
ap-7678	93	13	its	its	PRON
ap-7678	93	14	kinetic	kinetic	ADJ
ap-7678	93	15	energy	energy	NOUN
ap-7678	93	16	counterpart	counterpart	NOUN
ap-7678	93	17	is	be	AUX
ap-7678	93	18	set	set	VERB
ap-7678	93	19	to	to	ADP
ap-7678	93	20	zero	zero	NUM
ap-7678	93	21	.	.	PUNCT
ap-7678	94	1	this	this	PRON
ap-7678	94	2	is	be	AUX
ap-7678	94	3	one	one	NUM
ap-7678	94	4	simple	simple	ADJ
ap-7678	94	5	evidence	evidence	NOUN
ap-7678	94	6	that	that	SCONJ
ap-7678	94	7	the	the	DET
ap-7678	94	8	mer	mer	PROPN
ap-7678	94	9	transformation	transformation	NOUN
ap-7678	94	10	regulates	regulate	VERB
ap-7678	94	11	singular	singular	ADJ
ap-7678	94	12	perturbation	perturbation	NOUN
ap-7678	94	13	expansions	expansion	NOUN
ap-7678	94	14	(	(	PUNCT
ap-7678	94	15	i.e.	i.e.	X
ap-7678	94	16	kinetic	kinetic	ADJ
ap-7678	94	17	energy	energy	NOUN
ap-7678	94	18	expansions	expansion	NOUN
ap-7678	94	19	)	)	PUNCT
ap-7678	94	20	.	.	PUNCT
ap-7678	95	1	that	that	PRON
ap-7678	95	2	is	be	AUX
ap-7678	95	3	,	,	PUNCT
ap-7678	95	4	singular	singular	ADJ
ap-7678	95	5	perturbation	perturbation	NOUN
ap-7678	95	6	expansions	expansion	NOUN
ap-7678	95	7	in	in	ADP
ap-7678	95	8	the	the	DET
ap-7678	95	9	moments	moment	NOUN
ap-7678	95	10	’	'	PUNCT
ap-7678	95	11	representation	representation	NOUN
ap-7678	95	12	are	be	AUX
ap-7678	95	13	better	well	ADV
ap-7678	95	14	behaved	behave	VERB
ap-7678	95	15	.	.	PUNCT
ap-7678	96	1	3	3	X
ap-7678	96	2	.	.	X
ap-7678	96	3	generating	generate	VERB
ap-7678	96	4	eigenenergy	eigenenergy	NOUN
ap-7678	96	5	bounds	bound	NOUN
ap-7678	96	6	within	within	ADP
ap-7678	96	7	a	a	DET
ap-7678	96	8	moments	moment	NOUN
ap-7678	96	9	’	'	PUNCT
ap-7678	96	10	representation	representation	NOUN
ap-7678	96	11	there	there	PRON
ap-7678	96	12	are	be	VERB
ap-7678	96	13	two	two	NUM
ap-7678	96	14	methods	method	NOUN
ap-7678	96	15	for	for	ADP
ap-7678	96	16	generating	generate	VERB
ap-7678	96	17	bounds	bound	NOUN
ap-7678	96	18	within	within	ADP
ap-7678	96	19	a	a	DET
ap-7678	96	20	moment	moment	NOUN
ap-7678	96	21	equation	equation	NOUN
ap-7678	96	22	representation	representation	NOUN
ap-7678	96	23	(	(	PUNCT
ap-7678	96	24	mer	mer	NOUN
ap-7678	96	25	)	)	PUNCT
ap-7678	96	26	.	.	PUNCT
ap-7678	97	1	the	the	DET
ap-7678	97	2	first	first	ADJ
ap-7678	97	3	method	method	NOUN
ap-7678	97	4	,	,	PUNCT
ap-7678	97	5	referred	refer	VERB
ap-7678	97	6	to	to	ADP
ap-7678	97	7	as	as	ADP
ap-7678	97	8	the	the	DET
ap-7678	97	9	eigenvalue	eigenvalue	PROPN
ap-7678	97	10	moment	moment	NOUN
ap-7678	97	11	method	method	NOUN
ap-7678	97	12	(	(	PUNCT
ap-7678	97	13	emm	emm	PROPN
ap-7678	97	14	)	)	PUNCT
ap-7678	97	15	,	,	PUNCT
ap-7678	97	16	was	be	AUX
ap-7678	97	17	developed	develop	VERB
ap-7678	97	18	by	by	ADP
ap-7678	97	19	handy	handy	ADJ
ap-7678	97	20	and	and	CCONJ
ap-7678	97	21	bessis	bessis	NOUN
ap-7678	97	22	[	[	X
ap-7678	97	23	14	14	NUM
ap-7678	97	24	,	,	PUNCT
ap-7678	97	25	15	15	NUM
ap-7678	97	26	]	]	PUNCT
ap-7678	97	27	,	,	PUNCT
ap-7678	97	28	and	and	CCONJ
ap-7678	97	29	is	be	AUX
ap-7678	97	30	based	base	VERB
ap-7678	97	31	on	on	ADP
ap-7678	97	32	the	the	DET
ap-7678	97	33	moment	moment	NOUN
ap-7678	97	34	problem	problem	NOUN
ap-7678	98	1	[	[	X
ap-7678	98	2	13	13	NUM
ap-7678	98	3	]	]	PUNCT
ap-7678	98	4	.	.	PUNCT
ap-7678	99	1	its	its	PRON
ap-7678	99	2	theoretical	theoretical	ADJ
ap-7678	99	3	-	-	PUNCT
ap-7678	99	4	computational	computational	ADJ
ap-7678	99	5	structure	structure	NOUN
ap-7678	99	6	is	be	AUX
ap-7678	99	7	based	base	VERB
ap-7678	99	8	on	on	ADP
ap-7678	99	9	what	what	PRON
ap-7678	99	10	is	be	AUX
ap-7678	99	11	now	now	ADV
ap-7678	99	12	referred	refer	VERB
ap-7678	99	13	to	to	ADP
ap-7678	99	14	as	as	ADP
ap-7678	99	15	semidefnite	semidefnite	ADJ
ap-7678	99	16	programming	programming	NOUN
ap-7678	99	17	(	(	PUNCT
ap-7678	99	18	sdp	sdp	NOUN
ap-7678	99	19	)	)	PUNCT
ap-7678	100	1	[	[	X
ap-7678	100	2	21	21	NUM
ap-7678	100	3	,	,	PUNCT
ap-7678	100	4	22	22	NUM
ap-7678	100	5	]	]	PUNCT
ap-7678	100	6	.	.	PUNCT
ap-7678	101	1	as	as	ADP
ap-7678	101	2	such	such	ADJ
ap-7678	101	3	,	,	PUNCT
ap-7678	101	4	the	the	DET
ap-7678	101	5	sdp	sdp	NOUN
ap-7678	101	6	based	base	VERB
ap-7678	101	7	formulation	formulation	NOUN
ap-7678	101	8	of	of	ADP
ap-7678	101	9	handy	handy	ADJ
ap-7678	101	10	and	and	CCONJ
ap-7678	101	11	bessis	bessis	NOUN
ap-7678	101	12	is	be	AUX
ap-7678	101	13	the	the	DET
ap-7678	101	14	first	first	ADJ
ap-7678	101	15	use	use	NOUN
ap-7678	101	16	of	of	ADP
ap-7678	101	17	such	such	ADJ
ap-7678	101	18	methods	method	NOUN
ap-7678	101	19	for	for	ADP
ap-7678	101	20	quantum	quantum	NOUN
ap-7678	101	21	operators	operator	NOUN
ap-7678	101	22	[	[	X
ap-7678	101	23	22	22	NUM
ap-7678	101	24	]	]	PUNCT
ap-7678	101	25	.	.	PUNCT
ap-7678	102	1	its	its	PRON
ap-7678	102	2	computational	computational	ADJ
ap-7678	102	3	implementation	implementation	NOUN
ap-7678	102	4	was	be	AUX
ap-7678	102	5	done	do	VERB
ap-7678	102	6	through	through	ADP
ap-7678	102	7	the	the	DET
ap-7678	102	8	use	use	NOUN
ap-7678	102	9	of	of	ADP
ap-7678	102	10	linear	linear	ADJ
ap-7678	102	11	programming	programming	NOUN
ap-7678	102	12	[	[	X
ap-7678	102	13	23	23	NUM
ap-7678	102	14	]	]	PUNCT
ap-7678	102	15	,	,	PUNCT
ap-7678	102	16	since	since	SCONJ
ap-7678	102	17	sdp	sdp	NOUN
ap-7678	102	18	algorithms	algorithm	NOUN
ap-7678	102	19	were	be	AUX
ap-7678	102	20	not	not	PART
ap-7678	102	21	known	know	VERB
ap-7678	102	22	in	in	ADP
ap-7678	102	23	the	the	DET
ap-7678	102	24	1980s	1980s	NUM
ap-7678	102	25	.	.	PUNCT
ap-7678	103	1	the	the	DET
ap-7678	103	2	second	second	ADJ
ap-7678	103	3	moment	moment	NOUN
ap-7678	103	4	representation	representation	NOUN
ap-7678	103	5	bounding	bounding	NOUN
ap-7678	103	6	formulation	formulation	NOUN
ap-7678	103	7	is	be	AUX
ap-7678	103	8	that	that	SCONJ
ap-7678	103	9	presented	present	VERB
ap-7678	103	10	in	in	ADP
ap-7678	103	11	this	this	DET
ap-7678	103	12	work	work	NOUN
ap-7678	103	13	,	,	PUNCT
ap-7678	103	14	oppq	oppq	PROPN
ap-7678	103	15	-	-	PUNCT
ap-7678	103	16	bm	bm	PROPN
ap-7678	103	17	.	.	PUNCT
ap-7678	104	1	unlike	unlike	ADP
ap-7678	104	2	emm	emm	PROPN
ap-7678	104	3	,	,	PUNCT
ap-7678	104	4	oppq	oppq	PROPN
ap-7678	104	5	-	-	PUNCT
ap-7678	104	6	bm	bm	PROPN
ap-7678	104	7	is	be	AUX
ap-7678	104	8	applicable	applicable	ADJ
ap-7678	104	9	to	to	ADP
ap-7678	104	10	the	the	DET
ap-7678	104	11	low	low	ADJ
ap-7678	104	12	lying	lie	VERB
ap-7678	104	13	discrete	discrete	ADJ
ap-7678	104	14	states	state	NOUN
ap-7678	104	15	of	of	ADP
ap-7678	104	16	any	any	DET
ap-7678	104	17	system	system	NOUN
ap-7678	104	18	,	,	PUNCT
ap-7678	104	19	hermitian	hermitian	NOUN
ap-7678	104	20	,	,	PUNCT
ap-7678	104	21	or	or	CCONJ
ap-7678	104	22	nonhermitian	nonhermitian	ADJ
ap-7678	104	23	,	,	PUNCT
ap-7678	104	24	bosonic	bosonic	NOUN
ap-7678	104	25	,	,	PUNCT
ap-7678	104	26	or	or	CCONJ
ap-7678	104	27	fermionic	fermionic	NOUN
ap-7678	104	28	,	,	PUNCT
ap-7678	104	29	provided	provide	VERB
ap-7678	104	30	it	it	PRON
ap-7678	104	31	admits	admit	VERB
ap-7678	104	32	a	a	DET
ap-7678	104	33	moment	moment	NOUN
ap-7678	104	34	equation	equation	NOUN
ap-7678	104	35	representation	representation	NOUN
ap-7678	104	36	(	(	PUNCT
ap-7678	104	37	mer	mer	NOUN
ap-7678	104	38	)	)	PUNCT
ap-7678	104	39	.	.	PUNCT
ap-7678	105	1	emm	emm	PROPN
ap-7678	105	2	is	be	AUX
ap-7678	105	3	applicable	applicable	ADJ
ap-7678	105	4	only	only	ADV
ap-7678	105	5	for	for	ADP
ap-7678	105	6	the	the	DET
ap-7678	105	7	multidimensional	multidimensional	ADJ
ap-7678	105	8	bosonic	bosonic	NOUN
ap-7678	105	9	ground	ground	NOUN
ap-7678	105	10	state	state	NOUN
ap-7678	105	11	.	.	PUNCT
ap-7678	106	1	for	for	ADP
ap-7678	106	2	systems	system	NOUN
ap-7678	106	3	admitting	admit	VERB
ap-7678	106	4	both	both	DET
ap-7678	106	5	emm	emm	PROPN
ap-7678	106	6	and	and	CCONJ
ap-7678	106	7	oppq	oppq	PROPN
ap-7678	106	8	-	-	PUNCT
ap-7678	106	9	bm	bm	PROPN
ap-7678	106	10	,	,	PUNCT
ap-7678	106	11	it	it	PRON
ap-7678	106	12	is	be	AUX
ap-7678	106	13	our	our	PRON
ap-7678	106	14	belief	belief	NOUN
ap-7678	106	15	that	that	SCONJ
ap-7678	106	16	at	at	ADP
ap-7678	106	17	its	its	PRON
ap-7678	106	18	basic	basic	ADJ
ap-7678	106	19	level	level	NOUN
ap-7678	106	20	,	,	PUNCT
ap-7678	106	21	emm	emm	PROPN
ap-7678	106	22	produces	produce	VERB
ap-7678	106	23	faster	fast	ADV
ap-7678	106	24	converging	converge	VERB
ap-7678	106	25	bounds	bound	NOUN
ap-7678	106	26	(	(	PUNCT
ap-7678	106	27	as	as	SCONJ
ap-7678	106	28	shown	show	VERB
ap-7678	106	29	in	in	ADP
ap-7678	106	30	this	this	DET
ap-7678	106	31	work	work	NOUN
ap-7678	106	32	through	through	ADP
ap-7678	106	33	the	the	DET
ap-7678	106	34	analysis	analysis	NOUN
ap-7678	106	35	of	of	ADP
ap-7678	106	36	the	the	DET
ap-7678	106	37	sextic	sextic	ADJ
ap-7678	106	38	anharmonic	anharmonic	ADJ
ap-7678	106	39	oscillator	oscillator	NOUN
ap-7678	106	40	in	in	ADP
ap-7678	106	41	eq	eq	ADP
ap-7678	106	42	.	.	PUNCT
ap-7678	107	1	(	(	PUNCT
ap-7678	107	2	6	6	NUM
ap-7678	107	3	)	)	PUNCT
ap-7678	107	4	)	)	PUNCT
ap-7678	107	5	;	;	PUNCT
ap-7678	107	6	however	however	ADV
ap-7678	107	7	,	,	PUNCT
ap-7678	107	8	if	if	SCONJ
ap-7678	107	9	one	one	PRON
ap-7678	107	10	optimizes	optimize	VERB
ap-7678	107	11	the	the	DET
ap-7678	107	12	selection	selection	NOUN
ap-7678	107	13	of	of	ADP
ap-7678	107	14	the	the	DET
ap-7678	107	15	reference	reference	NOUN
ap-7678	107	16	/	/	SYM
ap-7678	107	17	weight	weight	NOUN
ap-7678	107	18	function	function	NOUN
ap-7678	107	19	,	,	PUNCT
ap-7678	107	20	then	then	ADV
ap-7678	107	21	oppq	oppq	NOUN
ap-7678	107	22	-	-	PUNCT
ap-7678	107	23	bm	bm	PROPN
ap-7678	107	24	can	can	AUX
ap-7678	107	25	yield	yield	VERB
ap-7678	107	26	significantly	significantly	ADV
ap-7678	107	27	faster	fast	ADV
ap-7678	107	28	converging	converge	VERB
ap-7678	107	29	results	result	NOUN
ap-7678	107	30	.	.	PUNCT
ap-7678	108	1	the	the	DET
ap-7678	108	2	emm	emm	PROPN
ap-7678	108	3	formulation	formulation	NOUN
ap-7678	108	4	involves	involve	VERB
ap-7678	108	5	sophisticated	sophisticated	ADJ
ap-7678	108	6	,	,	PUNCT
ap-7678	108	7	nonlinear	nonlinear	ADJ
ap-7678	108	8	,	,	PUNCT
ap-7678	108	9	convex	convex	ADJ
ap-7678	108	10	optimization	optimization	NOUN
ap-7678	108	11	analytical	analytical	ADJ
ap-7678	108	12	tools	tool	NOUN
ap-7678	108	13	.	.	PUNCT
ap-7678	109	1	however	however	ADV
ap-7678	109	2	,	,	PUNCT
ap-7678	109	3	oppq	oppq	PROPN
ap-7678	109	4	-	-	PUNCT
ap-7678	109	5	bm	bm	PROPN
ap-7678	109	6	is	be	AUX
ap-7678	109	7	purely	purely	ADV
ap-7678	109	8	algebraic	algebraic	ADJ
ap-7678	109	9	(	(	PUNCT
ap-7678	109	10	i.e.	i.e.	X
ap-7678	109	11	eigenvalues	eigenvalue	NOUN
ap-7678	109	12	,	,	PUNCT
ap-7678	109	13	eigenvectors	eigenvector	NOUN
ap-7678	109	14	,	,	PUNCT
ap-7678	109	15	and	and	CCONJ
ap-7678	109	16	algebra	algebra	NOUN
ap-7678	109	17	)	)	PUNCT
ap-7678	109	18	.	.	PUNCT
ap-7678	110	1	given	give	VERB
ap-7678	110	2	the	the	DET
ap-7678	110	3	power	power	NOUN
ap-7678	110	4	of	of	ADP
ap-7678	110	5	mathematica	mathematica	PROPN
ap-7678	110	6	,	,	PUNCT
ap-7678	110	7	with	with	ADP
ap-7678	110	8	unlimited	unlimited	ADJ
ap-7678	110	9	precision	precision	NOUN
ap-7678	110	10	,	,	PUNCT
ap-7678	110	11	it	it	PRON
ap-7678	110	12	can	can	AUX
ap-7678	110	13	produce	produce	VERB
ap-7678	110	14	spectacular	spectacular	ADJ
ap-7678	110	15	results	result	NOUN
ap-7678	110	16	.	.	PUNCT
ap-7678	111	1	it	it	PRON
ap-7678	111	2	is	be	AUX
ap-7678	111	3	important	important	ADJ
ap-7678	111	4	to	to	PART
ap-7678	111	5	stress	stress	VERB
ap-7678	111	6	that	that	SCONJ
ap-7678	111	7	emm	emm	PROPN
ap-7678	111	8	produces	produce	VERB
ap-7678	111	9	(	(	PUNCT
ap-7678	111	10	when	when	SCONJ
ap-7678	111	11	applicable	applicable	ADJ
ap-7678	111	12	)	)	PUNCT
ap-7678	111	13	tight	tight	ADJ
ap-7678	111	14	bounds	bound	NOUN
ap-7678	111	15	ab	ab	PROPN
ap-7678	111	16	initio	initio	PROPN
ap-7678	111	17	.	.	PUNCT
ap-7678	112	1	the	the	DET
ap-7678	112	2	oppq	oppq	PROPN
ap-7678	112	3	-	-	PUNCT
ap-7678	112	4	bm	bm	PROPN
ap-7678	112	5	is	be	AUX
ap-7678	112	6	empirical	empirical	ADJ
ap-7678	112	7	.	.	PUNCT
ap-7678	113	1	if	if	SCONJ
ap-7678	113	2	the	the	DET
ap-7678	113	3	convergence	convergence	NOUN
ap-7678	113	4	of	of	ADP
ap-7678	113	5	a	a	DET
ap-7678	113	6	particular	particular	ADJ
ap-7678	113	7	parameter	parameter	NOUN
ap-7678	113	8	is	be	AUX
ap-7678	113	9	numerically	numerically	ADV
ap-7678	113	10	observed	observe	VERB
ap-7678	113	11	,	,	PUNCT
ap-7678	113	12	then	then	ADV
ap-7678	113	13	one	one	PRON
ap-7678	113	14	can	can	AUX
ap-7678	113	15	confidently	confidently	ADV
ap-7678	113	16	generate	generate	VERB
ap-7678	113	17	bounds	bound	NOUN
ap-7678	113	18	for	for	ADP
ap-7678	113	19	the	the	DET
ap-7678	113	20	physical	physical	ADJ
ap-7678	113	21	energies	energy	NOUN
ap-7678	113	22	.	.	PUNCT
ap-7678	114	1	we	we	PRON
ap-7678	114	2	demonstrate	demonstrate	VERB
ap-7678	114	3	this	this	PRON
ap-7678	114	4	.	.	PUNCT
ap-7678	115	1	4	4	X
ap-7678	115	2	.	.	X
ap-7678	115	3	the	the	DET
ap-7678	115	4	eigenvalue	eigenvalue	PROPN
ap-7678	115	5	moment	moment	NOUN
ap-7678	115	6	method	method	VERB
ap-7678	115	7	the	the	DET
ap-7678	115	8	bosonic	bosonic	PROPN
ap-7678	115	9	ground	ground	NOUN
ap-7678	115	10	state	state	NOUN
ap-7678	115	11	must	must	AUX
ap-7678	115	12	be	be	AUX
ap-7678	115	13	a	a	DET
ap-7678	115	14	positive	positive	ADJ
ap-7678	115	15	(	(	PUNCT
ap-7678	115	16	nonnegative	nonnegative	ADJ
ap-7678	115	17	)	)	PUNCT
ap-7678	115	18	configuration	configuration	NOUN
ap-7678	116	1	[	[	X
ap-7678	116	2	24	24	NUM
ap-7678	116	3	]	]	PUNCT
ap-7678	116	4	,	,	PUNCT
ap-7678	116	5	ψgr(−→r	ψgr(−→r	PROPN
ap-7678	116	6	)	)	PUNCT
ap-7678	116	7	≥	≥	NOUN
ap-7678	116	8	0	0	NUM
ap-7678	116	9	.	.	PUNCT
ap-7678	117	1	if	if	SCONJ
ap-7678	117	2	the	the	DET
ap-7678	117	3	corresponding	correspond	VERB
ap-7678	117	4	schrödinger	schrödinger	ADJ
ap-7678	117	5	equation	equation	NOUN
ap-7678	117	6	is	be	AUX
ap-7678	117	7	transformable	transformable	ADJ
ap-7678	117	8	into	into	ADP
ap-7678	117	9	mer	mer	NOUN
ap-7678	117	10	form	form	NOUN
ap-7678	117	11	,	,	PUNCT
ap-7678	117	12	then	then	ADV
ap-7678	117	13	one	one	PRON
ap-7678	117	14	can	can	AUX
ap-7678	117	15	impose	impose	VERB
ap-7678	117	16	the	the	DET
ap-7678	117	17	moment	moment	NOUN
ap-7678	117	18	problem	problem	NOUN
ap-7678	117	19	positivity	positivity	NOUN
ap-7678	117	20	theorems	theorem	VERB
ap-7678	117	21	and	and	CCONJ
ap-7678	117	22	constrain	constrain	VERB
ap-7678	117	23	the	the	DET
ap-7678	117	24	power	power	NOUN
ap-7678	117	25	moments	moment	NOUN
ap-7678	117	26	,	,	PUNCT
ap-7678	117	27	and	and	CCONJ
ap-7678	117	28	in	in	ADP
ap-7678	117	29	turn	turn	NOUN
ap-7678	117	30	the	the	DET
ap-7678	117	31	energy	energy	NOUN
ap-7678	117	32	and	and	CCONJ
ap-7678	117	33	missing	missing	ADJ
ap-7678	117	34	moments	moment	NOUN
ap-7678	117	35	.	.	PUNCT
ap-7678	118	1	this	this	PRON
ap-7678	118	2	is	be	AUX
ap-7678	118	3	the	the	DET
ap-7678	118	4	emm	emm	PROPN
ap-7678	118	5	-	-	PUNCT
ap-7678	118	6	methodology	methodology	NOUN
ap-7678	118	7	.	.	PUNCT
ap-7678	119	1	this	this	PRON
ap-7678	119	2	was	be	AUX
ap-7678	119	3	done	do	VERB
ap-7678	119	4	,	,	PUNCT
ap-7678	119	5	several	several	ADJ
ap-7678	119	6	decades	decade	NOUN
ap-7678	119	7	ago	ago	ADV
ap-7678	119	8	,	,	PUNCT
ap-7678	119	9	by	by	ADP
ap-7678	119	10	handy	handy	ADJ
ap-7678	119	11	and	and	CCONJ
ap-7678	119	12	bessis	bessis	NOUN
ap-7678	119	13	(	(	PUNCT
ap-7678	119	14	hb	hb	X
ap-7678	119	15	)	)	PUNCT
ap-7678	120	1	[	[	X
ap-7678	120	2	14	14	NUM
ap-7678	120	3	,	,	PUNCT
ap-7678	120	4	15	15	NUM
ap-7678	120	5	]	]	PUNCT
ap-7678	120	6	.	.	PUNCT
ap-7678	121	1	the	the	DET
ap-7678	121	2	eigenvalue	eigenvalue	PROPN
ap-7678	121	3	moment	moment	NOUN
ap-7678	121	4	method	method	NOUN
ap-7678	121	5	(	(	PUNCT
ap-7678	121	6	emm	emm	PROPN
ap-7678	121	7	)	)	PUNCT
ap-7678	121	8	,	,	PUNCT
ap-7678	121	9	achieves	achieve	VERB
ap-7678	121	10	geometric	geometric	ADJ
ap-7678	121	11	convergence	convergence	NOUN
ap-7678	121	12	rates	rate	NOUN
ap-7678	121	13	for	for	ADP
ap-7678	121	14	the	the	DET
ap-7678	121	15	bounds	bound	NOUN
ap-7678	121	16	to	to	ADP
ap-7678	121	17	the	the	DET
ap-7678	121	18	ground	ground	NOUN
ap-7678	121	19	state	state	NOUN
ap-7678	121	20	energy	energy	NOUN
ap-7678	121	21	.	.	PUNCT
ap-7678	122	1	the	the	DET
ap-7678	122	2	only	only	ADJ
ap-7678	122	3	limitation	limitation	NOUN
ap-7678	122	4	is	be	AUX
ap-7678	122	5	that	that	SCONJ
ap-7678	122	6	it	it	PRON
ap-7678	122	7	can	can	AUX
ap-7678	122	8	only	only	ADV
ap-7678	122	9	be	be	AUX
ap-7678	122	10	applied	apply	VERB
ap-7678	122	11	to	to	ADP
ap-7678	122	12	multidimensional	multidimensional	ADJ
ap-7678	122	13	bosonic	bosonic	NOUN
ap-7678	122	14	systems	system	NOUN
ap-7678	122	15	,	,	PUNCT
ap-7678	122	16	and	and	CCONJ
ap-7678	122	17	then	then	ADV
ap-7678	122	18	only	only	ADV
ap-7678	122	19	to	to	ADP
ap-7678	122	20	the	the	DET
ap-7678	122	21	ground	ground	NOUN
ap-7678	122	22	state	state	NOUN
ap-7678	122	23	.	.	PUNCT
ap-7678	123	1	as	as	SCONJ
ap-7678	123	2	previously	previously	ADV
ap-7678	123	3	noted	note	VERB
ap-7678	123	4	,	,	PUNCT
ap-7678	123	5	it	it	PRON
ap-7678	123	6	was	be	AUX
ap-7678	123	7	used	use	VERB
ap-7678	123	8	to	to	PART
ap-7678	123	9	solve	solve	VERB
ap-7678	123	10	the	the	DET
ap-7678	123	11	quadratic	quadratic	ADJ
ap-7678	123	12	zeeman	zeeman	NOUN
ap-7678	123	13	(	(	PUNCT
ap-7678	123	14	qzm	qzm	PROPN
ap-7678	123	15	)	)	PUNCT
ap-7678	123	16	problem	problem	NOUN
ap-7678	123	17	[	[	X
ap-7678	123	18	15	15	NUM
ap-7678	123	19	]	]	PUNCT
ap-7678	123	20	.	.	PUNCT
ap-7678	124	1	the	the	DET
ap-7678	124	2	bounding	bounding	NOUN
ap-7678	124	3	of	of	ADP
ap-7678	124	4	bosonic	bosonic	NOUN
ap-7678	124	5	excited	excite	VERB
ap-7678	124	6	states	state	NOUN
ap-7678	124	7	,	,	PUNCT
ap-7678	124	8	through	through	ADP
ap-7678	124	9	a	a	DET
ap-7678	124	10	moment	moment	NOUN
ap-7678	124	11	representation	representation	NOUN
ap-7678	124	12	,	,	PUNCT
ap-7678	124	13	could	could	AUX
ap-7678	124	14	only	only	ADV
ap-7678	124	15	be	be	AUX
ap-7678	124	16	realized	realize	VERB
ap-7678	124	17	more	more	ADV
ap-7678	124	18	recently	recently	ADV
ap-7678	124	19	through	through	ADP
ap-7678	124	20	application	application	NOUN
ap-7678	124	21	of	of	ADP
ap-7678	124	22	oppq	oppq	NOUN
ap-7678	124	23	-	-	PUNCT
ap-7678	124	24	bm	bm	PROPN
ap-7678	125	1	[	[	X
ap-7678	125	2	9	9	NUM
ap-7678	125	3	]	]	PUNCT
ap-7678	125	4	.	.	PUNCT
ap-7678	126	1	one	one	PRON
ap-7678	126	2	can	can	AUX
ap-7678	126	3	extend	extend	VERB
ap-7678	126	4	the	the	DET
ap-7678	126	5	emm	emm	PROPN
ap-7678	126	6	quantization	quantization	NOUN
ap-7678	126	7	philosophy	philosophy	NOUN
ap-7678	126	8	to	to	ADP
ap-7678	126	9	the	the	DET
ap-7678	126	10	probability	probability	NOUN
ap-7678	126	11	density	density	NOUN
ap-7678	126	12	of	of	ADP
ap-7678	126	13	one	one	NUM
ap-7678	126	14	dimensional	dimensional	ADJ
ap-7678	126	15	hermitian	hermitian	ADJ
ap-7678	126	16	schrödinger	schrödinger	ADJ
ap-7678	126	17	operators	operator	NOUN
ap-7678	126	18	.	.	PUNCT
ap-7678	127	1	this	this	PRON
ap-7678	127	2	is	be	AUX
ap-7678	127	3	because	because	SCONJ
ap-7678	127	4	the	the	DET
ap-7678	127	5	probability	probability	NOUN
ap-7678	127	6	density	density	NOUN
ap-7678	127	7	will	will	AUX
ap-7678	127	8	satisfy	satisfy	VERB
ap-7678	127	9	a	a	DET
ap-7678	127	10	third	third	ADJ
ap-7678	127	11	order	order	NOUN
ap-7678	127	12	,	,	PUNCT
ap-7678	127	13	linear	linear	ADJ
ap-7678	127	14	,	,	PUNCT
ap-7678	127	15	differential	differential	ADJ
ap-7678	127	16	equation	equation	NOUN
ap-7678	127	17	(	(	PUNCT
ap-7678	127	18	lode	lode	PROPN
ap-7678	127	19	)	)	PUNCT
ap-7678	127	20	.	.	PUNCT
ap-7678	128	1	if	if	SCONJ
ap-7678	128	2	the	the	DET
ap-7678	128	3	lode	lode	PROPN
ap-7678	128	4	representation	representation	NOUN
ap-7678	128	5	admits	admit	VERB
ap-7678	128	6	a	a	DET
ap-7678	128	7	mer	mer	PROPN
ap-7678	128	8	formulation	formulation	NOUN
ap-7678	128	9	,	,	PUNCT
ap-7678	128	10	then	then	ADV
ap-7678	128	11	one	one	PRON
ap-7678	128	12	can	can	AUX
ap-7678	128	13	generate	generate	VERB
ap-7678	128	14	tight	tight	ADJ
ap-7678	128	15	bounds	bound	NOUN
ap-7678	128	16	to	to	ADP
ap-7678	128	17	any	any	DET
ap-7678	128	18	discrete	discrete	ADJ
ap-7678	128	19	state	state	NOUN
ap-7678	128	20	.	.	PUNCT
ap-7678	129	1	this	this	PRON
ap-7678	129	2	is	be	AUX
ap-7678	129	3	because	because	SCONJ
ap-7678	129	4	all	all	DET
ap-7678	129	5	the	the	DET
ap-7678	129	6	discrete	discrete	ADJ
ap-7678	129	7	states	state	NOUN
ap-7678	129	8	are	be	AUX
ap-7678	129	9	associated	associate	VERB
ap-7678	129	10	with	with	ADP
ap-7678	129	11	nonnegative	nonnegative	ADJ
ap-7678	129	12	,	,	PUNCT
ap-7678	129	13	l2	l2	NOUN
ap-7678	129	14	,	,	PUNCT
ap-7678	129	15	configurations	configuration	NOUN
ap-7678	129	16	.	.	PUNCT
ap-7678	130	1	if	if	SCONJ
ap-7678	130	2	the	the	DET
ap-7678	130	3	underlying	underlie	VERB
ap-7678	130	4	sturm	sturm	NOUN
ap-7678	130	5	-	-	PUNCT
ap-7678	130	6	liouville	liouville	NOUN
ap-7678	130	7	problem	problem	NOUN
ap-7678	130	8	is	be	AUX
ap-7678	130	9	nonhermitian	nonhermitian	ADJ
ap-7678	130	10	,	,	PUNCT
ap-7678	130	11	then	then	ADV
ap-7678	130	12	the	the	DET
ap-7678	130	13	one	one	NUM
ap-7678	130	14	dimensional	dimensional	ADJ
ap-7678	130	15	schrödinger	schrödinger	ADJ
ap-7678	130	16	equation	equation	NOUN
ap-7678	130	17	can	can	AUX
ap-7678	130	18	be	be	AUX
ap-7678	130	19	transformed	transform	VERB
ap-7678	130	20	into	into	ADP
ap-7678	130	21	a	a	DET
ap-7678	130	22	fourth	fourth	ADJ
ap-7678	130	23	order	order	NOUN
ap-7678	130	24	lode	lode	NOUN
ap-7678	130	25	for	for	ADP
ap-7678	130	26	the	the	DET
ap-7678	130	27	probability	probability	NOUN
ap-7678	130	28	density	density	NOUN
ap-7678	130	29	.	.	PUNCT
ap-7678	131	1	this	this	DET
ap-7678	131	2	approach	approach	NOUN
ap-7678	131	3	was	be	AUX
ap-7678	131	4	used	use	VERB
ap-7678	131	5	by	by	ADP
ap-7678	131	6	handy	handy	ADJ
ap-7678	131	7	[	[	X
ap-7678	131	8	8	8	NUM
ap-7678	131	9	]	]	PUNCT
ap-7678	131	10	in	in	ADP
ap-7678	131	11	precisely	precisely	ADV
ap-7678	131	12	computing	compute	VERB
ap-7678	131	13	the	the	DET
ap-7678	131	14	a	a	DET
ap-7678	131	15	-	-	PUNCT
ap-7678	131	16	parameter	parameter	NOUN
ap-7678	131	17	regimes	regime	NOUN
ap-7678	131	18	where	where	SCONJ
ap-7678	131	19	the	the	DET
ap-7678	131	20	system	system	NOUN
ap-7678	131	21	v	v	NOUN
ap-7678	131	22	(	(	PUNCT
ap-7678	131	23	x	x	NOUN
ap-7678	131	24	)	)	PUNCT
ap-7678	131	25	=	=	SYM
ap-7678	131	26	ix3	ix3	PROPN
ap-7678	131	27	+	+	CCONJ
ap-7678	131	28	iax	iax	PROPN
ap-7678	131	29	,	,	PUNCT
ap-7678	131	30	violated	violate	VERB
ap-7678	131	31	pt	pt	NOUN
ap-7678	131	32	symmetry	symmetry	NOUN
ap-7678	131	33	(	(	PUNCT
ap-7678	131	34	i.e.	i.e.	X
ap-7678	131	35	pt	pt	X
ap-7678	131	36	symmetry	symmetry	NOUN
ap-7678	131	37	breaking	breaking	NOUN
ap-7678	131	38	)	)	PUNCT
ap-7678	131	39	.	.	PUNCT
ap-7678	132	1	these	these	DET
ap-7678	132	2	results	result	NOUN
ap-7678	132	3	were	be	AUX
ap-7678	132	4	confirmed	confirm	VERB
ap-7678	132	5	through	through	ADP
ap-7678	132	6	a	a	DET
ap-7678	132	7	faster	fast	ADJ
ap-7678	132	8	,	,	PUNCT
ap-7678	132	9	moment	moment	NOUN
ap-7678	132	10	based	base	VERB
ap-7678	132	11	,	,	PUNCT
ap-7678	132	12	estimation	estimation	NOUN
ap-7678	132	13	procedure	procedure	NOUN
ap-7678	132	14	[	[	X
ap-7678	132	15	25	25	NUM
ap-7678	132	16	]	]	PUNCT
ap-7678	132	17	,	,	PUNCT
ap-7678	132	18	that	that	PRON
ap-7678	132	19	led	lead	VERB
ap-7678	132	20	to	to	ADP
ap-7678	132	21	a	a	DET
ap-7678	132	22	more	more	ADV
ap-7678	132	23	detailed	detailed	ADJ
ap-7678	132	24	understanding	understanding	NOUN
ap-7678	132	25	for	for	ADP
ap-7678	132	26	the	the	DET
ap-7678	132	27	onset	onset	NOUN
ap-7678	132	28	of	of	ADP
ap-7678	132	29	pt	pt	NOUN
ap-7678	132	30	symmetry	symmetry	NOUN
ap-7678	132	31	breaking	breaking	NOUN
ap-7678	132	32	.	.	PUNCT
ap-7678	133	1	these	these	DET
ap-7678	133	2	results	result	NOUN
ap-7678	133	3	significantly	significantly	ADV
ap-7678	133	4	improved	improve	VERB
ap-7678	133	5	upon	upon	SCONJ
ap-7678	133	6	the	the	DET
ap-7678	133	7	predictions	prediction	NOUN
ap-7678	133	8	by	by	ADP
ap-7678	133	9	delabaere	delabaere	ADV
ap-7678	133	10	and	and	CCONJ
ap-7678	133	11	trinh	trinh	PROPN
ap-7678	134	1	[	[	X
ap-7678	134	2	26	26	NUM
ap-7678	134	3	]	]	PUNCT
ap-7678	134	4	,	,	PUNCT
ap-7678	134	5	based	base	VERB
ap-7678	134	6	on	on	ADP
ap-7678	134	7	asymptotic	asymptotic	ADJ
ap-7678	134	8	analysis	analysis	NOUN
ap-7678	134	9	.	.	PUNCT
ap-7678	135	1	we	we	PRON
ap-7678	135	2	revisit	revisit	VERB
ap-7678	135	3	this	this	DET
ap-7678	135	4	problem	problem	NOUN
ap-7678	135	5	in	in	ADP
ap-7678	135	6	this	this	DET
ap-7678	135	7	work	work	NOUN
ap-7678	135	8	.	.	PUNCT
ap-7678	136	1	the	the	DET
ap-7678	136	2	multidimensional	multidimensional	ADJ
ap-7678	136	3	probability	probability	NOUN
ap-7678	136	4	density	density	NOUN
ap-7678	136	5	,	,	PUNCT
ap-7678	136	6	s(−→r	s(−→r	PROPN
ap-7678	136	7	)	)	PUNCT
ap-7678	136	8	≡	≡	PROPN
ap-7678	137	1	ψ∗(−→r	ψ∗(−→r	NOUN
ap-7678	137	2	)	)	PUNCT
ap-7678	137	3	ψ(−→r	ψ(−→r	PROPN
ap-7678	137	4	)	)	PUNCT
ap-7678	137	5	will	will	AUX
ap-7678	137	6	not	not	PART
ap-7678	137	7	generally	generally	ADV
ap-7678	137	8	satisfy	satisfy	VERB
ap-7678	137	9	a	a	DET
ap-7678	137	10	linear	linear	ADJ
ap-7678	137	11	partial	partial	ADJ
ap-7678	137	12	differential	differential	NOUN
ap-7678	137	13	equation	equation	NOUN
ap-7678	137	14	;	;	PUNCT
ap-7678	137	15	therefore	therefore	ADV
ap-7678	137	16	,	,	PUNCT
ap-7678	137	17	no	no	DET
ap-7678	137	18	mer	mer	PROPN
ap-7678	137	19	relation	relation	NOUN
ap-7678	137	20	can	can	AUX
ap-7678	137	21	be	be	AUX
ap-7678	137	22	generated	generate	VERB
ap-7678	137	23	,	,	PUNCT
ap-7678	137	24	and	and	CCONJ
ap-7678	137	25	emm	emm	PROPN
ap-7678	137	26	can	can	AUX
ap-7678	137	27	not	not	PART
ap-7678	137	28	be	be	AUX
ap-7678	137	29	applied	apply	VERB
ap-7678	137	30	.	.	PUNCT
ap-7678	138	1	nevertheless	nevertheless	ADV
ap-7678	138	2	,	,	PUNCT
ap-7678	138	3	through	through	ADP
ap-7678	138	4	oppq	oppq	NOUN
ap-7678	138	5	-	-	PUNCT
ap-7678	138	6	bm	bm	VERB
ap-7678	138	7	we	we	PRON
ap-7678	138	8	can	can	AUX
ap-7678	138	9	circumvent	circumvent	VERB
ap-7678	138	10	this	this	DET
ap-7678	138	11	difficulty	difficulty	NOUN
ap-7678	138	12	.	.	PUNCT
ap-7678	139	1	65	65	NUM
ap-7678	139	2	carlos	carlos	PROPN
ap-7678	139	3	r.	r.	PROPN
ap-7678	139	4	handy	handy	PROPN
ap-7678	139	5	acta	acta	PROPN
ap-7678	139	6	polytechnica	polytechnica	PROPN
ap-7678	139	7	5	5	NUM
ap-7678	139	8	.	.	PUNCT
ap-7678	140	1	the	the	DET
ap-7678	140	2	orthonormal	orthonormal	ADJ
ap-7678	140	3	polynomial	polynomial	ADJ
ap-7678	140	4	projection	projection	NOUN
ap-7678	140	5	quantization	quantization	NOUN
ap-7678	140	6	formalism	formalism	NOUN
ap-7678	140	7	5.1	5.1	NUM
ap-7678	140	8	.	.	PUNCT
ap-7678	141	1	the	the	DET
ap-7678	141	2	oppq	oppq	PROPN
ap-7678	141	3	non	non	ADJ
ap-7678	141	4	-	-	ADJ
ap-7678	141	5	orthogonal	orthogonal	ADJ
ap-7678	141	6	basis	basis	NOUN
ap-7678	141	7	expansion	expansion	NOUN
ap-7678	141	8	let	let	VERB
ap-7678	141	9	us	we	PRON
ap-7678	141	10	expand	expand	VERB
ap-7678	141	11	the	the	DET
ap-7678	141	12	discrete	discrete	ADJ
ap-7678	141	13	state	state	NOUN
ap-7678	141	14	wavefunction	wavefunction	NOUN
ap-7678	141	15	in	in	ADP
ap-7678	141	16	terms	term	NOUN
ap-7678	141	17	of	of	ADP
ap-7678	141	18	some	some	DET
ap-7678	141	19	appropriate	appropriate	ADJ
ap-7678	141	20	,	,	PUNCT
ap-7678	141	21	complete	complete	ADJ
ap-7678	141	22	,	,	PUNCT
ap-7678	141	23	non	non	ADJ
ap-7678	141	24	-	-	ADJ
ap-7678	141	25	orthonormal	orthonormal	ADJ
ap-7678	141	26	basis	basis	NOUN
ap-7678	141	27	,	,	PUNCT
ap-7678	141	28	{	{	PUNCT
ap-7678	141	29	bn(x	bn(x	X
ap-7678	141	30	)	)	PUNCT
ap-7678	141	31	≡	≡	PROPN
ap-7678	141	32	pn(x)r(x	pn(x)r(x	PROPN
ap-7678	141	33	)	)	PUNCT
ap-7678	141	34	}	}	PUNCT
ap-7678	141	35	:	:	PUNCT
ap-7678	141	36	ψ(x	ψ(x	X
ap-7678	141	37	)	)	PUNCT
ap-7678	142	1	=	=	PUNCT
ap-7678	143	1	∞∑	∞∑	PRON
ap-7678	143	2	n=0	n=0	NUM
ap-7678	143	3	cnpn(x)r(x	cnpn(x)r(x	NOUN
ap-7678	143	4	)	)	PUNCT
ap-7678	143	5	.	.	PUNCT
ap-7678	144	1	(	(	PUNCT
ap-7678	144	2	9	9	X
ap-7678	144	3	)	)	PUNCT
ap-7678	144	4	we	we	PRON
ap-7678	144	5	require	require	VERB
ap-7678	144	6	a	a	DET
ap-7678	144	7	weighted	weight	VERB
ap-7678	144	8	polynomial	polynomial	ADJ
ap-7678	144	9	basis	basis	NOUN
ap-7678	144	10	,	,	PUNCT
ap-7678	144	11	involving	involve	VERB
ap-7678	144	12	orthonormal	orthonormal	ADJ
ap-7678	144	13	polynomials	polynomial	NOUN
ap-7678	144	14	relative	relative	ADJ
ap-7678	144	15	to	to	ADP
ap-7678	144	16	some	some	DET
ap-7678	144	17	appropriate	appropriate	ADJ
ap-7678	144	18	positive	positive	ADJ
ap-7678	144	19	weight	weight	NOUN
ap-7678	144	20	r(x	r(x	NOUN
ap-7678	144	21	)	)	PUNCT
ap-7678	144	22	>	>	X
ap-7678	144	23	0	0	PUNCT
ap-7678	145	1	(	(	PUNCT
ap-7678	145	2	we	we	PRON
ap-7678	145	3	adopt	adopt	VERB
ap-7678	145	4	the	the	DET
ap-7678	145	5	one	one	NUM
ap-7678	145	6	dimensional	dimensional	ADJ
ap-7678	145	7	notation	notation	NOUN
ap-7678	145	8	for	for	ADP
ap-7678	145	9	simplicity	simplicity	NOUN
ap-7678	145	10	):	):	PUNCT
ap-7678	145	11	⟨pm|r|pn⟩	⟨pm|r|pn⟩	NOUN
ap-7678	145	12	=	=	SYM
ap-7678	145	13	δm	δm	PROPN
ap-7678	145	14	,	,	PUNCT
ap-7678	145	15	n.	n.	NOUN
ap-7678	145	16	(	(	PUNCT
ap-7678	145	17	10	10	NUM
ap-7678	145	18	)	)	PUNCT
ap-7678	145	19	the	the	DET
ap-7678	145	20	weight	weight	NOUN
ap-7678	145	21	is	be	AUX
ap-7678	145	22	real	real	ADJ
ap-7678	145	23	,	,	PUNCT
ap-7678	145	24	as	as	SCONJ
ap-7678	145	25	are	be	AUX
ap-7678	145	26	its	its	PRON
ap-7678	145	27	orthonormal	orthonormal	ADJ
ap-7678	145	28	polynomials	polynomial	NOUN
ap-7678	145	29	,	,	PUNCT
ap-7678	145	30	involving	involve	VERB
ap-7678	145	31	real	real	ADJ
ap-7678	145	32	polynomial	polynomial	ADJ
ap-7678	145	33	coefficients	coefficient	NOUN
ap-7678	145	34	:	:	PUNCT
ap-7678	145	35	pn(x	pn(x	X
ap-7678	145	36	)	)	PUNCT
ap-7678	145	37	=	=	SYM
ap-7678	145	38	n∑	n∑	NOUN
ap-7678	145	39	j=0	j=0	PROPN
ap-7678	145	40	ξ(n	ξ(n	PROPN
ap-7678	145	41	)	)	PUNCT
ap-7678	145	42	j	j	PROPN
ap-7678	145	43	xj	xj	PROPN
ap-7678	145	44	.	.	PUNCT
ap-7678	146	1	(	(	PUNCT
ap-7678	146	2	11	11	NUM
ap-7678	146	3	)	)	PUNCT
ap-7678	146	4	for	for	ADP
ap-7678	146	5	non	non	ADJ
ap-7678	146	6	-	-	ADJ
ap-7678	146	7	hermitian	hermitian	ADJ
ap-7678	146	8	systems	system	NOUN
ap-7678	146	9	with	with	ADP
ap-7678	146	10	complex	complex	ADJ
ap-7678	146	11	bound	bind	VERB
ap-7678	146	12	state	state	NOUN
ap-7678	146	13	wavefunctions	wavefunction	NOUN
ap-7678	146	14	,	,	PUNCT
ap-7678	146	15	it	it	PRON
ap-7678	146	16	is	be	AUX
ap-7678	146	17	the	the	DET
ap-7678	146	18	projection	projection	NOUN
ap-7678	146	19	coefficients	coefficient	NOUN
ap-7678	146	20	,	,	PUNCT
ap-7678	146	21	{	{	PUNCT
ap-7678	146	22	cn	cn	NOUN
ap-7678	146	23	}	}	PUNCT
ap-7678	146	24	that	that	PRON
ap-7678	146	25	become	become	VERB
ap-7678	146	26	complex	complex	ADJ
ap-7678	146	27	.	.	PUNCT
ap-7678	147	1	the	the	DET
ap-7678	147	2	basis	basis	NOUN
ap-7678	147	3	{	{	PUNCT
ap-7678	147	4	bn(x)r(x	bn(x)r(x	PROPN
ap-7678	147	5	)	)	PUNCT
ap-7678	147	6	}	}	PUNCT
ap-7678	147	7	will	will	AUX
ap-7678	147	8	be	be	AUX
ap-7678	147	9	complete	complete	ADJ
ap-7678	147	10	,	,	PUNCT
ap-7678	147	11	but	but	CCONJ
ap-7678	147	12	nonorthogonal	nonorthogonal	ADJ
ap-7678	147	13	(	(	PUNCT
ap-7678	147	14	i.e.	i.e.	X
ap-7678	147	15	⟨bm|bn⟩	⟨bm|bn⟩	X
ap-7678	147	16	=	=	SYM
ap-7678	147	17	̸	̸	NUM
ap-7678	147	18	0	0	NUM
ap-7678	147	19	,	,	PUNCT
ap-7678	147	20	if	if	SCONJ
ap-7678	147	21	m	m	PROPN
ap-7678	147	22	̸=	̸=	PROPN
ap-7678	147	23	n	n	CCONJ
ap-7678	147	24	)	)	PUNCT
ap-7678	147	25	.	.	PUNCT
ap-7678	148	1	we	we	PRON
ap-7678	148	2	can	can	AUX
ap-7678	148	3	rewrite	rewrite	VERB
ap-7678	148	4	eq	eq	ADP
ap-7678	148	5	.	.	PUNCT
ap-7678	149	1	(	(	PUNCT
ap-7678	149	2	10	10	NUM
ap-7678	149	3	)	)	PUNCT
ap-7678	149	4	as	as	ADP
ap-7678	149	5	m∑	m∑	CCONJ
ap-7678	149	6	j1=0	j1=0	PROPN
ap-7678	149	7	n∑	n∑	PROPN
ap-7678	149	8	j2=0	j2=0	PROPN
ap-7678	150	1	ξ(m	ξ(m	NOUN
ap-7678	150	2	)	)	PUNCT
ap-7678	150	3	j1	j1	NOUN
ap-7678	150	4	ω(j1	ω(j1	PUNCT
ap-7678	151	1	+	+	SYM
ap-7678	151	2	j2)ξ(n	j2)ξ(n	PROPN
ap-7678	151	3	)	)	PUNCT
ap-7678	151	4	j2	j2	NOUN
ap-7678	151	5	=	=	PUNCT
ap-7678	151	6	δm	δm	PROPN
ap-7678	151	7	,	,	PUNCT
ap-7678	151	8	n	n	CCONJ
ap-7678	151	9	,	,	PUNCT
ap-7678	151	10	(	(	PUNCT
ap-7678	151	11	12	12	NUM
ap-7678	151	12	)	)	PUNCT
ap-7678	151	13	where	where	SCONJ
ap-7678	151	14	ω(j1	ω(j1	NOUN
ap-7678	152	1	+	+	ADJ
ap-7678	152	2	j2	j2	NOUN
ap-7678	152	3	)	)	PUNCT
ap-7678	152	4	=	=	SYM
ap-7678	153	1	∫	∫	PROPN
ap-7678	153	2	dx	dx	PROPN
ap-7678	153	3	xj1+j2r(x	xj1+j2r(x	PROPN
ap-7678	153	4	)	)	PUNCT
ap-7678	153	5	,	,	PUNCT
ap-7678	153	6	the	the	DET
ap-7678	153	7	hankel	hankel	NOUN
ap-7678	153	8	moment	moment	NOUN
ap-7678	153	9	matrix	matrix	NOUN
ap-7678	153	10	of	of	ADP
ap-7678	153	11	the	the	DET
ap-7678	153	12	weight	weight	NOUN
ap-7678	153	13	,	,	PUNCT
ap-7678	153	14	wi	wi	PROPN
ap-7678	153	15	,	,	PUNCT
ap-7678	153	16	j	j	PROPN
ap-7678	153	17	≡	≡	PROPN
ap-7678	153	18	ω(i	ω(i	PROPN
ap-7678	153	19	+	+	CCONJ
ap-7678	153	20	j	j	PROPN
ap-7678	153	21	)	)	PUNCT
ap-7678	153	22	.	.	PUNCT
ap-7678	154	1	knowledge	knowledge	NOUN
ap-7678	154	2	of	of	ADP
ap-7678	154	3	the	the	DET
ap-7678	154	4	hankel	hankel	NOUN
ap-7678	154	5	moment	moment	NOUN
ap-7678	154	6	matrix	matrix	NOUN
ap-7678	154	7	allows	allow	VERB
ap-7678	154	8	us	we	PRON
ap-7678	154	9	to	to	PART
ap-7678	154	10	generate	generate	VERB
ap-7678	154	11	the	the	DET
ap-7678	154	12	orthonormal	orthonormal	ADJ
ap-7678	154	13	polynomials	polynomial	NOUN
ap-7678	154	14	through	through	ADP
ap-7678	154	15	the	the	DET
ap-7678	154	16	cholesky	cholesky	ADJ
ap-7678	154	17	decomposition	decomposition	NOUN
ap-7678	154	18	method	method	NOUN
ap-7678	154	19	,	,	PUNCT
ap-7678	154	20	which	which	PRON
ap-7678	154	21	involves	involve	VERB
ap-7678	154	22	decomposing	decompose	VERB
ap-7678	154	23	the	the	DET
ap-7678	154	24	positive	positive	ADJ
ap-7678	154	25	hankel	hankel	NOUN
ap-7678	154	26	matrix	matrix	NOUN
ap-7678	154	27	into	into	ADP
ap-7678	154	28	the	the	DET
ap-7678	154	29	form	form	NOUN
ap-7678	154	30	w	w	NOUN
ap-7678	154	31	=	=	PUNCT
ap-7678	154	32	cc†.	cc†.	NOUN
ap-7678	154	33	let	let	VERB
ap-7678	154	34	êj	êj	PART
ap-7678	154	35	correspond	correspond	VERB
ap-7678	154	36	to	to	ADP
ap-7678	154	37	a	a	DET
ap-7678	154	38	unit	unit	NOUN
ap-7678	154	39	vector	vector	NOUN
ap-7678	154	40	in	in	ADP
ap-7678	154	41	the	the	DET
ap-7678	154	42	j	j	PROPN
ap-7678	154	43	-	-	PUNCT
ap-7678	154	44	th	th	VERB
ap-7678	154	45	component	component	NOUN
ap-7678	154	46	.	.	PUNCT
ap-7678	155	1	we	we	PRON
ap-7678	155	2	then	then	ADV
ap-7678	155	3	solve	solve	VERB
ap-7678	155	4	for	for	ADP
ap-7678	155	5	−→ξ	−→ξ	X
ap-7678	155	6	(	(	PUNCT
ap-7678	155	7	j	j	NOUN
ap-7678	155	8	)	)	PUNCT
ap-7678	155	9	=	=	SYM
ap-7678	156	1	(	(	PUNCT
ap-7678	156	2	c†)−1êj	c†)−1êj	PROPN
ap-7678	156	3	.	.	PUNCT
ap-7678	157	1	(	(	PUNCT
ap-7678	157	2	13	13	NUM
ap-7678	157	3	)	)	PUNCT
ap-7678	157	4	this	this	PRON
ap-7678	157	5	generates	generate	VERB
ap-7678	157	6	the	the	DET
ap-7678	157	7	coefficient	coefficient	NOUN
ap-7678	157	8	vector	vector	NOUN
ap-7678	157	9	for	for	ADP
ap-7678	157	10	pn(x	pn(x	NOUN
ap-7678	157	11	)	)	PUNCT
ap-7678	157	12	,	,	PUNCT
ap-7678	157	13	or	or	CCONJ
ap-7678	157	14	−−→	−−→	PROPN
ap-7678	157	15	ξ(n	ξ(n	PROPN
ap-7678	157	16	)	)	PUNCT
ap-7678	157	17	.	.	PUNCT
ap-7678	158	1	the	the	DET
ap-7678	158	2	projection	projection	NOUN
ap-7678	158	3	coefficients	coefficient	NOUN
ap-7678	158	4	are	be	AUX
ap-7678	158	5	obtainable	obtainable	ADJ
ap-7678	158	6	from	from	ADP
ap-7678	158	7	the	the	DET
ap-7678	158	8	mer	mer	PROPN
ap-7678	158	9	relation	relation	NOUN
ap-7678	158	10	for	for	ADP
ap-7678	158	11	the	the	DET
ap-7678	158	12	power	power	NOUN
ap-7678	158	13	moments	moment	NOUN
ap-7678	158	14	of	of	ADP
ap-7678	158	15	ψ(x	ψ(x	NOUN
ap-7678	158	16	):	):	PUNCT
ap-7678	158	17	µ(p	µ(p	PROPN
ap-7678	158	18	)	)	PUNCT
ap-7678	158	19	≡	≡	PROPN
ap-7678	158	20	∫	∫	PROPN
ap-7678	158	21	ℜ	ℜ	PROPN
ap-7678	158	22	dx	dx	PROPN
ap-7678	158	23	xpψ(x	xpψ(x	PROPN
ap-7678	158	24	)	)	PUNCT
ap-7678	158	25	.	.	PUNCT
ap-7678	159	1	(	(	PUNCT
ap-7678	159	2	14	14	NUM
ap-7678	159	3	)	)	PUNCT
ap-7678	159	4	assume	assume	VERB
ap-7678	159	5	that	that	SCONJ
ap-7678	159	6	the	the	DET
ap-7678	159	7	corresponding	correspond	VERB
ap-7678	159	8	mer	mer	PROPN
ap-7678	159	9	relation	relation	NOUN
ap-7678	159	10	exists	exist	VERB
ap-7678	159	11	µ(p	µ(p	ADJ
ap-7678	159	12	)	)	PUNCT
ap-7678	159	13	=	=	SYM
ap-7678	159	14	ms∑	ms∑	X
ap-7678	159	15	ℓ=0	ℓ=0	SYM
ap-7678	159	16	me(p	me(p	NOUN
ap-7678	159	17	,	,	PUNCT
ap-7678	159	18	ℓ)µℓ.	ℓ)µℓ.	NUM
ap-7678	159	19	(	(	PUNCT
ap-7678	159	20	15	15	NUM
ap-7678	159	21	)	)	PUNCT
ap-7678	159	22	it	it	PRON
ap-7678	159	23	then	then	ADV
ap-7678	159	24	follows	follow	VERB
ap-7678	159	25	that	that	SCONJ
ap-7678	159	26	cn	cn	PROPN
ap-7678	159	27	=	=	SYM
ap-7678	159	28	⟨pn|ψ⟩	⟨pn|ψ⟩	PROPN
ap-7678	159	29	,	,	PUNCT
ap-7678	159	30	=	=	SYM
ap-7678	159	31	n∑	n∑	NOUN
ap-7678	159	32	j=0	j=0	PROPN
ap-7678	159	33	ξ(n	ξ(n	PROPN
ap-7678	159	34	)	)	PUNCT
ap-7678	159	35	j	j	PROPN
ap-7678	159	36	µ(j	µ(j	PROPN
ap-7678	159	37	)	)	PUNCT
ap-7678	160	1	=	=	SYM
ap-7678	160	2	n∑	n∑	NOUN
ap-7678	160	3	j=0	j=0	PROPN
ap-7678	160	4	ξ(n	ξ(n	PROPN
ap-7678	160	5	)	)	PUNCT
ap-7678	160	6	j	j	PROPN
ap-7678	160	7	ms∑	ms∑	PROPN
ap-7678	160	8	ℓ=0	ℓ=0	PROPN
ap-7678	160	9	me(j	me(j	NOUN
ap-7678	160	10	,	,	PUNCT
ap-7678	160	11	ℓ)µℓ	ℓ)µℓ	PROPN
ap-7678	160	12	,	,	PUNCT
ap-7678	160	13	(	(	PUNCT
ap-7678	160	14	16	16	NUM
ap-7678	160	15	)	)	PUNCT
ap-7678	160	16	cn(−→µ	cn(−→µ	NOUN
ap-7678	160	17	)	)	PUNCT
ap-7678	161	1	=	=	SYM
ap-7678	161	2	ms∑	ms∑	X
ap-7678	161	3	ℓ=0	ℓ=0	SYM
ap-7678	161	4	λ(n	λ(n	NOUN
ap-7678	161	5	)	)	PUNCT
ap-7678	161	6	ℓ	ℓ	PROPN
ap-7678	161	7	(	(	PUNCT
ap-7678	161	8	e)µℓ	e)µℓ	PROPN
ap-7678	161	9	,	,	PUNCT
ap-7678	161	10	(	(	PUNCT
ap-7678	161	11	17	17	NUM
ap-7678	161	12	)	)	PUNCT
ap-7678	161	13	where	where	SCONJ
ap-7678	161	14	λ(n	λ(n	PROPN
ap-7678	161	15	)	)	PUNCT
ap-7678	161	16	ℓ	ℓ	PROPN
ap-7678	161	17	(	(	PUNCT
ap-7678	161	18	e	e	NOUN
ap-7678	161	19	)	)	PUNCT
ap-7678	161	20	=	=	SYM
ap-7678	161	21	n∑	n∑	NOUN
ap-7678	161	22	j=0	j=0	PROPN
ap-7678	161	23	ξ(n	ξ(n	PROPN
ap-7678	161	24	)	)	PUNCT
ap-7678	161	25	j	j	PROPN
ap-7678	161	26	me(j	me(j	NOUN
ap-7678	161	27	,	,	PUNCT
ap-7678	161	28	ℓ	ℓ	NOUN
ap-7678	161	29	)	)	PUNCT
ap-7678	161	30	.	.	PUNCT
ap-7678	162	1	(	(	PUNCT
ap-7678	162	2	18	18	NUM
ap-7678	162	3	)	)	PUNCT
ap-7678	162	4	the	the	DET
ap-7678	162	5	mer	mer	PROPN
ap-7678	162	6	relation	relation	PROPN
ap-7678	162	7	suggested	suggest	VERB
ap-7678	162	8	in	in	ADP
ap-7678	162	9	eq	eq	ADP
ap-7678	162	10	.	.	PUNCT
ap-7678	163	1	(	(	PUNCT
ap-7678	163	2	15	15	NUM
ap-7678	163	3	)	)	PUNCT
ap-7678	163	4	is	be	AUX
ap-7678	163	5	a	a	DET
ap-7678	163	6	homogeneous	homogeneous	ADJ
ap-7678	163	7	relation	relation	NOUN
ap-7678	163	8	for	for	ADP
ap-7678	163	9	the	the	DET
ap-7678	163	10	power	power	NOUN
ap-7678	163	11	moments	moment	NOUN
ap-7678	163	12	.	.	PUNCT
ap-7678	164	1	some	some	DET
ap-7678	164	2	normalization	normalization	NOUN
ap-7678	164	3	condition	condition	NOUN
ap-7678	164	4	needs	need	VERB
ap-7678	164	5	to	to	PART
ap-7678	164	6	be	be	AUX
ap-7678	164	7	imposed	impose	VERB
ap-7678	164	8	:	:	PUNCT
ap-7678	164	9	c(−→µ	c(−→µ	NOUN
ap-7678	164	10	)	)	PUNCT
ap-7678	164	11	=	=	SYM
ap-7678	165	1	1	1	X
ap-7678	165	2	.	.	PUNCT
ap-7678	165	3	(	(	PUNCT
ap-7678	165	4	19	19	NUM
ap-7678	165	5	)	)	PUNCT
ap-7678	165	6	for	for	ADP
ap-7678	165	7	one	one	NUM
ap-7678	165	8	dimensional	dimensional	ADJ
ap-7678	165	9	systems	system	NOUN
ap-7678	165	10	,	,	PUNCT
ap-7678	165	11	the	the	DET
ap-7678	165	12	natural	natural	ADJ
ap-7678	165	13	normalization	normalization	NOUN
ap-7678	165	14	is	be	AUX
ap-7678	165	15	the	the	DET
ap-7678	165	16	unit	unit	NOUN
ap-7678	165	17	(	(	PUNCT
ap-7678	165	18	nonlinear	nonlinear	ADJ
ap-7678	165	19	)	)	PUNCT
ap-7678	165	20	normalization	normalization	NOUN
ap-7678	165	21	:	:	PUNCT
ap-7678	165	22	|−→µ	|−→µ	NOUN
ap-7678	165	23	|2	|2	NUM
ap-7678	165	24	=	=	SYM
ap-7678	165	25	1	1	NUM
ap-7678	165	26	.	.	PUNCT
ap-7678	166	1	however	however	ADV
ap-7678	166	2	,	,	PUNCT
ap-7678	166	3	it	it	PRON
ap-7678	166	4	need	need	AUX
ap-7678	166	5	not	not	PART
ap-7678	166	6	be	be	AUX
ap-7678	166	7	chosen	choose	VERB
ap-7678	166	8	as	as	ADP
ap-7678	166	9	such	such	ADJ
ap-7678	166	10	.	.	PUNCT
ap-7678	167	1	alternative	alternative	ADJ
ap-7678	167	2	choices	choice	NOUN
ap-7678	167	3	[	[	X
ap-7678	167	4	9	9	NUM
ap-7678	167	5	]	]	PUNCT
ap-7678	167	6	are	be	AUX
ap-7678	167	7	linear	linear	ADJ
ap-7678	167	8	normalizations	normalization	NOUN
ap-7678	167	9	such	such	ADJ
ap-7678	167	10	as	as	ADP
ap-7678	167	11	µ0	µ0	NOUN
ap-7678	167	12	=	=	SYM
ap-7678	167	13	1	1	NUM
ap-7678	167	14	or	or	CCONJ
ap-7678	167	15	µ0	µ0	NOUN
ap-7678	167	16	+	+	CCONJ
ap-7678	167	17	µ1	µ1	NOUN
ap-7678	167	18	=	=	SYM
ap-7678	167	19	1	1	NUM
ap-7678	167	20	,	,	PUNCT
ap-7678	167	21	etc	etc	X
ap-7678	167	22	.	.	X
ap-7678	167	23	5.2	5.2	NUM
ap-7678	167	24	.	.	PUNCT
ap-7678	168	1	the	the	DET
ap-7678	168	2	oppq	oppq	ADJ
ap-7678	168	3	quantization	quantization	NOUN
ap-7678	168	4	condition	condition	NOUN
ap-7678	168	5	the	the	DET
ap-7678	168	6	oppq	oppq	ADJ
ap-7678	168	7	quantization	quantization	NOUN
ap-7678	168	8	condition	condition	NOUN
ap-7678	168	9	requires	require	VERB
ap-7678	168	10	that	that	SCONJ
ap-7678	168	11	the	the	DET
ap-7678	168	12	weight	weight	NOUN
ap-7678	168	13	be	be	AUX
ap-7678	168	14	chosen	choose	VERB
ap-7678	168	15	so	so	SCONJ
ap-7678	168	16	that	that	SCONJ
ap-7678	168	17	the	the	DET
ap-7678	168	18	following	follow	VERB
ap-7678	168	19	positive	positive	ADJ
ap-7678	168	20	integral	integral	NOUN
ap-7678	168	21	is	be	AUX
ap-7678	168	22	bounded	bound	VERB
ap-7678	168	23	for	for	ADP
ap-7678	168	24	discrete	discrete	ADJ
ap-7678	168	25	solutions	solution	NOUN
ap-7678	168	26	and	and	CCONJ
ap-7678	168	27	infinite	infinite	VERB
ap-7678	168	28	for	for	ADP
ap-7678	168	29	unphysical	unphysical	ADJ
ap-7678	168	30	solutions	solution	NOUN
ap-7678	168	31	:	:	PUNCT
ap-7678	168	32	i[ψ	i[ψ	NOUN
ap-7678	168	33	,	,	PUNCT
ap-7678	168	34	r	r	NOUN
ap-7678	168	35	]	]	PUNCT
ap-7678	168	36	=	=	PUNCT
ap-7678	168	37	∫	∫	PROPN
ap-7678	168	38	ℜ	ℜ	PROPN
ap-7678	168	39	dx	dx	PROPN
ap-7678	168	40	|ψ(x)|2	|ψ(x)|2	X
ap-7678	168	41	r(x	r(x	PROPN
ap-7678	168	42	)	)	PUNCT
ap-7678	168	43	,	,	PUNCT
ap-7678	168	44	(	(	PUNCT
ap-7678	168	45	20	20	NUM
ap-7678	168	46	)	)	PUNCT
ap-7678	168	47	=	=	NOUN
ap-7678	169	1	∞∑	∞∑	NUM
ap-7678	169	2	j=0	j=0	PROPN
ap-7678	169	3	|cj(e	|cj(e	PROPN
ap-7678	169	4	,	,	PUNCT
ap-7678	169	5	−→µ	−→µ	NOUN
ap-7678	169	6	)	)	PUNCT
ap-7678	169	7	|2	|2	NUM
ap-7678	169	8	,	,	PUNCT
ap-7678	169	9	(	(	PUNCT
ap-7678	169	10	21	21	NUM
ap-7678	169	11	)	)	PUNCT
ap-7678	169	12	where	where	SCONJ
ap-7678	169	13	we	we	PRON
ap-7678	169	14	have	have	AUX
ap-7678	169	15	assumed	assume	VERB
ap-7678	169	16	ψ	ψ	ADP
ap-7678	169	17	is	be	AUX
ap-7678	169	18	a	a	DET
ap-7678	169	19	bounded	bounded	ADJ
ap-7678	169	20	discrete	discrete	ADJ
ap-7678	169	21	state	state	NOUN
ap-7678	169	22	.	.	PUNCT
ap-7678	170	1	more	more	ADV
ap-7678	170	2	generally	generally	ADV
ap-7678	170	3	,	,	PUNCT
ap-7678	170	4	we	we	PRON
ap-7678	170	5	note	note	VERB
ap-7678	170	6	that	that	SCONJ
ap-7678	170	7	depending	depend	VERB
ap-7678	170	8	on	on	ADP
ap-7678	170	9	the	the	DET
ap-7678	170	10	asymptotic	asymptotic	ADJ
ap-7678	170	11	behavior	behavior	NOUN
ap-7678	170	12	of	of	ADP
ap-7678	170	13	the	the	DET
ap-7678	170	14	physical	physical	ADJ
ap-7678	170	15	or	or	CCONJ
ap-7678	170	16	unphysical	unphysical	ADJ
ap-7678	170	17	(	(	PUNCT
ap-7678	170	18	i.e.	i.e.	X
ap-7678	170	19	unbounded	unbounded	ADJ
ap-7678	170	20	)	)	PUNCT
ap-7678	170	21	solutions	solution	NOUN
ap-7678	170	22	,	,	PUNCT
ap-7678	170	23	and	and	CCONJ
ap-7678	170	24	the	the	DET
ap-7678	170	25	chosen	choose	VERB
ap-7678	170	26	asymptoic	asymptoic	ADJ
ap-7678	170	27	form	form	NOUN
ap-7678	170	28	for	for	ADP
ap-7678	170	29	the	the	DET
ap-7678	170	30	weight	weight	NOUN
ap-7678	170	31	,	,	PUNCT
ap-7678	170	32	the	the	DET
ap-7678	170	33	integral	integral	NOUN
ap-7678	170	34	in	in	ADP
ap-7678	170	35	eq	eq	PROPN
ap-7678	170	36	.	.	PUNCT
ap-7678	171	1	(	(	PUNCT
ap-7678	171	2	20	20	NUM
ap-7678	171	3	)	)	PUNCT
ap-7678	171	4	will	will	AUX
ap-7678	171	5	satisfy	satisfy	VERB
ap-7678	171	6	the	the	DET
ap-7678	171	7	quantization	quantization	NOUN
ap-7678	171	8	condition	condition	NOUN
ap-7678	171	9	:	:	PUNCT
ap-7678	171	10	i[ψ	i[ψ	NOUN
ap-7678	171	11	,	,	PUNCT
ap-7678	171	12	r	r	NOUN
ap-7678	171	13	]	]	X
ap-7678	171	14	=	=	SYM
ap-7678	171	15	{	{	PUNCT
ap-7678	171	16	finite	finite	PROPN
ap-7678	171	17	,	,	PUNCT
ap-7678	171	18	⇐	⇐	ADJ
ap-7678	171	19	⇒	⇒	NOUN
ap-7678	171	20	e	e	NOUN
ap-7678	171	21	=	=	SYM
ap-7678	171	22	ephys	ephy	NOUN
ap-7678	171	23	and	and	CCONJ
ap-7678	171	24	−→µ	−→µ	NOUN
ap-7678	171	25	=	=	SYM
ap-7678	171	26	−→µ	−→µ	NOUN
ap-7678	171	27	phys	phy	NOUN
ap-7678	171	28	∞	∞	PROPN
ap-7678	171	29	,	,	PUNCT
ap-7678	171	30	⇐	⇐	ADJ
ap-7678	171	31	⇒	⇒	NOUN
ap-7678	171	32	e	e	X
ap-7678	171	33	̸=	̸=	PROPN
ap-7678	171	34	ephys	ephy	NOUN
ap-7678	171	35	or	or	CCONJ
ap-7678	171	36	−→µ	−→µ	NOUN
ap-7678	171	37	̸=	̸=	PROPN
ap-7678	171	38	−→µ	−→µ	NOUN
ap-7678	171	39	phys	phy	NOUN
ap-7678	171	40	.	.	PUNCT
ap-7678	172	1	(	(	PUNCT
ap-7678	172	2	22	22	NUM
ap-7678	172	3	)	)	PUNCT
ap-7678	172	4	the	the	DET
ap-7678	172	5	oppq	oppq	ADJ
ap-7678	172	6	quantization	quantization	NOUN
ap-7678	172	7	condition	condition	NOUN
ap-7678	172	8	essentially	essentially	ADV
ap-7678	172	9	becomes	become	VERB
ap-7678	172	10	a	a	DET
ap-7678	172	11	shooting	shooting	NOUN
ap-7678	172	12	method	method	NOUN
ap-7678	172	13	in	in	ADP
ap-7678	172	14	the	the	DET
ap-7678	172	15	e	e	NOUN
ap-7678	172	16	×	×	NOUN
ap-7678	172	17	−→µ	−→µ	NOUN
ap-7678	172	18	,	,	PUNCT
ap-7678	172	19	1	1	NUM
ap-7678	173	1	+	+	CCONJ
ap-7678	173	2	ms	ms	ADJ
ap-7678	173	3	,	,	PUNCT
ap-7678	173	4	parameter	parameter	NOUN
ap-7678	173	5	space	space	NOUN
ap-7678	173	6	(	(	PUNCT
ap-7678	173	7	after	after	ADP
ap-7678	173	8	imposing	impose	VERB
ap-7678	173	9	a	a	DET
ap-7678	173	10	normalization	normalization	NOUN
ap-7678	173	11	)	)	PUNCT
ap-7678	173	12	.	.	PUNCT
ap-7678	174	1	this	this	PRON
ap-7678	174	2	is	be	AUX
ap-7678	174	3	the	the	DET
ap-7678	174	4	essence	essence	NOUN
ap-7678	174	5	of	of	ADP
ap-7678	174	6	the	the	DET
ap-7678	174	7	oppq	oppq	PROPN
ap-7678	174	8	-	-	PUNCT
ap-7678	174	9	bm	bm	PROPN
ap-7678	174	10	bounding	bounding	NOUN
ap-7678	174	11	procedure	procedure	NOUN
ap-7678	174	12	.	.	PUNCT
ap-7678	175	1	the	the	DET
ap-7678	175	2	focus	focus	NOUN
ap-7678	175	3	of	of	ADP
ap-7678	175	4	the	the	DET
ap-7678	175	5	remaining	remain	VERB
ap-7678	175	6	oppq	oppq	ADJ
ap-7678	175	7	formalism	formalism	NOUN
ap-7678	175	8	is	be	AUX
ap-7678	175	9	to	to	PART
ap-7678	175	10	reduce	reduce	VERB
ap-7678	175	11	this	this	DET
ap-7678	175	12	shooting	shooting	NOUN
ap-7678	175	13	method	method	NOUN
ap-7678	175	14	to	to	ADP
ap-7678	175	15	a	a	DET
ap-7678	175	16	minimization	minimization	NOUN
ap-7678	175	17	problem	problem	NOUN
ap-7678	175	18	in	in	ADP
ap-7678	175	19	the	the	DET
ap-7678	175	20	energy	energy	NOUN
ap-7678	175	21	parameter	parameter	NOUN
ap-7678	175	22	space	space	NOUN
ap-7678	175	23	.	.	PUNCT
ap-7678	176	1	66	66	NUM
ap-7678	176	2	vol	vol	NOUN
ap-7678	176	3	.	.	PUNCT
ap-7678	177	1	62	62	NUM
ap-7678	177	2	no	no	INTJ
ap-7678	177	3	.	.	PUNCT
ap-7678	178	1	1/2022	1/2022	NUM
ap-7678	178	2	algebraic	algebraic	ADJ
ap-7678	178	3	eigenenergy	eigenenergy	NOUN
ap-7678	178	4	bounding	bounding	NOUN
ap-7678	178	5	method	method	NOUN
ap-7678	178	6	5.2.1	5.2.1	NUM
ap-7678	178	7	.	.	PUNCT
ap-7678	178	8	selection	selection	NOUN
ap-7678	178	9	of	of	ADP
ap-7678	178	10	the	the	DET
ap-7678	178	11	oppq	oppq	ADJ
ap-7678	178	12	weight	weight	NOUN
ap-7678	178	13	the	the	DET
ap-7678	178	14	above	above	ADJ
ap-7678	178	15	formalism	formalism	NOUN
ap-7678	178	16	has	have	VERB
ap-7678	178	17	two	two	NUM
ap-7678	178	18	significant	significant	ADJ
ap-7678	178	19	advantages	advantage	NOUN
ap-7678	178	20	.	.	PUNCT
ap-7678	179	1	the	the	DET
ap-7678	179	2	first	first	ADJ
ap-7678	179	3	is	be	AUX
ap-7678	179	4	that	that	SCONJ
ap-7678	179	5	it	it	PRON
ap-7678	179	6	tells	tell	VERB
ap-7678	179	7	us	we	PRON
ap-7678	179	8	that	that	SCONJ
ap-7678	179	9	the	the	DET
ap-7678	179	10	weight	weight	NOUN
ap-7678	179	11	should	should	AUX
ap-7678	179	12	not	not	PART
ap-7678	179	13	be	be	AUX
ap-7678	179	14	chosen	choose	VERB
ap-7678	179	15	so	so	SCONJ
ap-7678	179	16	that	that	SCONJ
ap-7678	179	17	it	it	PRON
ap-7678	179	18	asymptotically	asymptotically	ADV
ap-7678	179	19	vanishes	vanish	VERB
ap-7678	179	20	much	much	ADV
ap-7678	179	21	faster	fast	ADV
ap-7678	179	22	than	than	ADP
ap-7678	179	23	the	the	DET
ap-7678	179	24	asymptotic	asymptotic	ADJ
ap-7678	179	25	form	form	NOUN
ap-7678	179	26	of	of	ADP
ap-7678	179	27	the	the	DET
ap-7678	179	28	physical	physical	ADJ
ap-7678	179	29	solutions	solution	NOUN
ap-7678	179	30	.	.	PUNCT
ap-7678	180	1	thus	thus	ADV
ap-7678	180	2	,	,	PUNCT
ap-7678	180	3	given	give	VERB
ap-7678	180	4	a	a	DET
ap-7678	180	5	physical	physical	ADJ
ap-7678	180	6	,	,	PUNCT
ap-7678	180	7	discrete	discrete	ADJ
ap-7678	180	8	state	state	NOUN
ap-7678	180	9	,	,	PUNCT
ap-7678	180	10	ψ	ψ	VERB
ap-7678	180	11	,	,	PUNCT
ap-7678	180	12	let	let	AUX
ap-7678	180	13	c(x	c(x	NOUN
ap-7678	180	14	)	)	PUNCT
ap-7678	180	15	satisfy	satisfy	VERB
ap-7678	180	16	the	the	DET
ap-7678	180	17	asymptotic	asymptotic	ADJ
ap-7678	180	18	relation	relation	NOUN
ap-7678	180	19	:	:	PUNCT
ap-7678	180	20	lim|x|→∞	lim|x|→∞	NOUN
ap-7678	180	21	ψ(x	ψ(x	NOUN
ap-7678	180	22	)	)	PUNCT
ap-7678	180	23	r(x	r(x	PROPN
ap-7678	180	24	)	)	PUNCT
ap-7678	181	1	=	=	SYM
ap-7678	181	2	c(x	c(x	NOUN
ap-7678	181	3	)	)	PUNCT
ap-7678	181	4	.	.	PUNCT
ap-7678	182	1	(	(	PUNCT
ap-7678	182	2	23	23	NUM
ap-7678	182	3	)	)	PUNCT
ap-7678	182	4	we	we	PRON
ap-7678	182	5	then	then	ADV
ap-7678	182	6	want	want	VERB
ap-7678	182	7	∫	∫	PROPN
ap-7678	182	8	dx|ψ(x)c(x)|	dx|ψ(x)c(x)|	PROPN
ap-7678	182	9	<	<	X
ap-7678	182	10	∞.	∞.	PROPN
ap-7678	182	11	(	(	PUNCT
ap-7678	182	12	24	24	NUM
ap-7678	182	13	)	)	PUNCT
ap-7678	182	14	in	in	ADP
ap-7678	182	15	the	the	DET
ap-7678	182	16	works	work	NOUN
ap-7678	182	17	by	by	ADP
ap-7678	182	18	handy	handy	ADJ
ap-7678	182	19	and	and	CCONJ
ap-7678	182	20	vrinceanu	vrinceanu	NOUN
ap-7678	183	1	[	[	X
ap-7678	183	2	10	10	NUM
ap-7678	183	3	,	,	PUNCT
ap-7678	183	4	11	11	NUM
ap-7678	183	5	]	]	PUNCT
ap-7678	183	6	,	,	PUNCT
ap-7678	183	7	which	which	PRON
ap-7678	183	8	introduce	introduce	VERB
ap-7678	183	9	a	a	DET
ap-7678	183	10	particular	particular	ADJ
ap-7678	183	11	version	version	NOUN
ap-7678	183	12	of	of	ADP
ap-7678	183	13	the	the	DET
ap-7678	183	14	oppq	oppq	ADJ
ap-7678	183	15	formalism	formalism	NOUN
ap-7678	183	16	(	(	PUNCT
ap-7678	183	17	i.e.	i.e.	X
ap-7678	183	18	to	to	PART
ap-7678	183	19	be	be	AUX
ap-7678	183	20	referred	refer	VERB
ap-7678	183	21	to	to	ADP
ap-7678	183	22	here	here	ADV
ap-7678	183	23	to	to	PART
ap-7678	183	24	as	as	ADP
ap-7678	183	25	the	the	DET
ap-7678	183	26	oppqapproximation	oppqapproximation	NOUN
ap-7678	183	27	method	method	NOUN
ap-7678	183	28	(	(	PUNCT
ap-7678	183	29	oppq	oppq	PROPN
ap-7678	183	30	-	-	PUNCT
ap-7678	183	31	am	am	NOUN
ap-7678	183	32	)	)	PUNCT
ap-7678	183	33	)	)	PUNCT
ap-7678	183	34	,	,	PUNCT
ap-7678	183	35	they	they	PRON
ap-7678	183	36	argued	argue	VERB
ap-7678	183	37	and	and	CCONJ
ap-7678	183	38	demonstrated	demonstrate	VERB
ap-7678	183	39	that	that	SCONJ
ap-7678	183	40	the	the	DET
ap-7678	183	41	fastest	fast	ADJ
ap-7678	183	42	convergence	convergence	NOUN
ap-7678	183	43	to	to	ADP
ap-7678	183	44	the	the	DET
ap-7678	183	45	discrete	discrete	ADJ
ap-7678	183	46	states	state	NOUN
ap-7678	183	47	is	be	AUX
ap-7678	183	48	associated	associate	VERB
ap-7678	183	49	with	with	ADP
ap-7678	183	50	weights	weight	NOUN
ap-7678	183	51	that	that	PRON
ap-7678	183	52	mimic	mimic	VERB
ap-7678	183	53	the	the	DET
ap-7678	183	54	asymptotic	asymptotic	ADJ
ap-7678	183	55	form	form	NOUN
ap-7678	183	56	of	of	ADP
ap-7678	183	57	the	the	DET
ap-7678	183	58	desired	desire	VERB
ap-7678	183	59	discrete	discrete	ADJ
ap-7678	183	60	state	state	NOUN
ap-7678	183	61	:	:	PUNCT
ap-7678	183	62	lim|x|→∞c(x	lim|x|→∞c(x	NOUN
ap-7678	183	63	)	)	PUNCT
ap-7678	183	64	=	=	SYM
ap-7678	184	1	const	const	ADJ
ap-7678	184	2	.	.	PUNCT
ap-7678	185	1	(	(	PUNCT
ap-7678	185	2	25	25	NUM
ap-7678	185	3	)	)	PUNCT
ap-7678	185	4	the	the	DET
ap-7678	185	5	constant	constant	ADJ
ap-7678	185	6	can	can	AUX
ap-7678	185	7	be	be	AUX
ap-7678	185	8	finite	finite	ADJ
ap-7678	185	9	(	(	PUNCT
ap-7678	185	10	i.e.	i.e.	X
ap-7678	185	11	zero	zero	NUM
ap-7678	185	12	)	)	PUNCT
ap-7678	185	13	,	,	PUNCT
ap-7678	185	14	but	but	CCONJ
ap-7678	185	15	not	not	PART
ap-7678	185	16	infinite	infinite	VERB
ap-7678	185	17	in	in	ADP
ap-7678	185	18	a	a	DET
ap-7678	185	19	manner	manner	NOUN
ap-7678	185	20	that	that	PRON
ap-7678	185	21	violates	violate	VERB
ap-7678	185	22	eq	eq	ADP
ap-7678	185	23	.	.	PUNCT
ap-7678	186	1	(	(	PUNCT
ap-7678	186	2	24	24	NUM
ap-7678	186	3	)	)	PUNCT
ap-7678	186	4	.	.	PUNCT
ap-7678	187	1	5.2.2	5.2.2	NUM
ap-7678	187	2	.	.	X
ap-7678	187	3	use	use	NOUN
ap-7678	187	4	of	of	ADP
ap-7678	187	5	the	the	DET
ap-7678	187	6	ground	ground	NOUN
ap-7678	187	7	state	state	NOUN
ap-7678	187	8	as	as	ADP
ap-7678	187	9	a	a	DET
ap-7678	187	10	weight	weight	NOUN
ap-7678	187	11	another	another	DET
ap-7678	187	12	advantage	advantage	NOUN
ap-7678	187	13	of	of	ADP
ap-7678	187	14	the	the	DET
ap-7678	187	15	above	above	ADJ
ap-7678	187	16	formalism	formalism	NOUN
ap-7678	187	17	is	be	AUX
ap-7678	187	18	that	that	SCONJ
ap-7678	187	19	we	we	PRON
ap-7678	187	20	can	can	AUX
ap-7678	187	21	take	take	VERB
ap-7678	187	22	r(x	r(x	NOUN
ap-7678	187	23	)	)	PUNCT
ap-7678	187	24	=	=	SYM
ap-7678	187	25	ψgr(x	ψgr(x	PROPN
ap-7678	187	26	)	)	PUNCT
ap-7678	187	27	even	even	ADV
ap-7678	187	28	if	if	SCONJ
ap-7678	187	29	we	we	PRON
ap-7678	187	30	do	do	AUX
ap-7678	187	31	not	not	PART
ap-7678	187	32	know	know	VERB
ap-7678	187	33	the	the	DET
ap-7678	187	34	functional	functional	ADJ
ap-7678	187	35	form	form	NOUN
ap-7678	187	36	for	for	ADP
ap-7678	187	37	the	the	DET
ap-7678	187	38	ground	ground	NOUN
ap-7678	187	39	state	state	NOUN
ap-7678	187	40	.	.	PUNCT
ap-7678	188	1	as	as	ADV
ap-7678	188	2	long	long	ADV
ap-7678	188	3	as	as	SCONJ
ap-7678	188	4	one	one	PRON
ap-7678	188	5	does	do	AUX
ap-7678	188	6	not	not	PART
ap-7678	188	7	require	require	VERB
ap-7678	188	8	that	that	SCONJ
ap-7678	188	9	the	the	DET
ap-7678	188	10	discrete	discrete	ADJ
ap-7678	188	11	state	state	NOUN
ap-7678	188	12	wavefunction	wavefunction	NOUN
ap-7678	188	13	be	be	AUX
ap-7678	188	14	reconstructed	reconstruct	VERB
ap-7678	188	15	(	(	PUNCT
ap-7678	188	16	i.e.	i.e.	X
ap-7678	188	17	one	one	NUM
ap-7678	188	18	is	be	AUX
ap-7678	188	19	only	only	ADV
ap-7678	188	20	interested	interested	ADJ
ap-7678	188	21	in	in	ADP
ap-7678	188	22	the	the	DET
ap-7678	188	23	eigenenergies	eigenenergie	NOUN
ap-7678	188	24	)	)	PUNCT
ap-7678	188	25	then	then	ADV
ap-7678	188	26	the	the	DET
ap-7678	188	27	only	only	ADJ
ap-7678	188	28	information	information	NOUN
ap-7678	188	29	required	require	VERB
ap-7678	188	30	for	for	ADP
ap-7678	188	31	the	the	DET
ap-7678	188	32	ground	ground	NOUN
ap-7678	188	33	state	state	NOUN
ap-7678	188	34	is	be	AUX
ap-7678	188	35	that	that	SCONJ
ap-7678	188	36	its	its	PRON
ap-7678	188	37	power	power	NOUN
ap-7678	188	38	moments	moment	NOUN
ap-7678	188	39	be	be	AUX
ap-7678	188	40	known	know	VERB
ap-7678	188	41	accurately	accurately	ADV
ap-7678	188	42	.	.	PUNCT
ap-7678	189	1	this	this	PRON
ap-7678	189	2	then	then	ADV
ap-7678	189	3	allows	allow	VERB
ap-7678	189	4	us	we	PRON
ap-7678	189	5	to	to	PART
ap-7678	189	6	generate	generate	VERB
ap-7678	189	7	the	the	DET
ap-7678	189	8	corresponding	corresponding	ADJ
ap-7678	189	9	orthonormal	orthonormal	ADJ
ap-7678	189	10	polynomials	polynomial	NOUN
ap-7678	189	11	,	,	PUNCT
ap-7678	189	12	allowing	allow	VERB
ap-7678	189	13	for	for	ADP
ap-7678	189	14	the	the	DET
ap-7678	189	15	generation	generation	NOUN
ap-7678	189	16	of	of	ADP
ap-7678	189	17	rapidly	rapidly	ADV
ap-7678	189	18	converging	converge	VERB
ap-7678	189	19	bounds	bound	NOUN
ap-7678	189	20	to	to	ADP
ap-7678	189	21	the	the	DET
ap-7678	189	22	discrete	discrete	ADJ
ap-7678	189	23	states	state	NOUN
ap-7678	189	24	.	.	PUNCT
ap-7678	190	1	we	we	PRON
ap-7678	190	2	demonstrate	demonstrate	VERB
ap-7678	190	3	this	this	PRON
ap-7678	190	4	in	in	ADP
ap-7678	190	5	this	this	DET
ap-7678	190	6	work	work	NOUN
ap-7678	190	7	(	(	PUNCT
ap-7678	190	8	i.e.	i.e.	X
ap-7678	190	9	table	table	NOUN
ap-7678	190	10	8)	8)	NUM
ap-7678	190	11	.	.	PUNCT
ap-7678	191	1	one	one	PRON
ap-7678	191	2	can	can	AUX
ap-7678	191	3	determine	determine	VERB
ap-7678	191	4	the	the	DET
ap-7678	191	5	power	power	NOUN
ap-7678	191	6	moments	moment	NOUN
ap-7678	191	7	of	of	ADP
ap-7678	191	8	the	the	DET
ap-7678	191	9	ground	ground	NOUN
ap-7678	191	10	state	state	NOUN
ap-7678	191	11	wavefunction	wavefunction	NOUN
ap-7678	191	12	either	either	CCONJ
ap-7678	191	13	through	through	ADP
ap-7678	191	14	emm	emm	PROPN
ap-7678	191	15	or	or	CCONJ
ap-7678	191	16	oppq	oppq	PROPN
ap-7678	191	17	-	-	PUNCT
ap-7678	191	18	bm	bm	PROPN
ap-7678	191	19	.	.	PUNCT
ap-7678	191	20	note	note	VERB
ap-7678	191	21	that	that	SCONJ
ap-7678	191	22	for	for	ADP
ap-7678	191	23	bosonic	bosonic	NOUN
ap-7678	191	24	systems	system	NOUN
ap-7678	191	25	,	,	PUNCT
ap-7678	191	26	this	this	PRON
ap-7678	191	27	is	be	AUX
ap-7678	191	28	an	an	DET
ap-7678	191	29	excellent	excellent	ADJ
ap-7678	191	30	use	use	NOUN
ap-7678	191	31	of	of	ADP
ap-7678	191	32	emm	emm	PROPN
ap-7678	191	33	,	,	PUNCT
ap-7678	191	34	since	since	SCONJ
ap-7678	191	35	the	the	DET
ap-7678	191	36	ground	ground	NOUN
ap-7678	191	37	state	state	NOUN
ap-7678	191	38	is	be	AUX
ap-7678	191	39	usually	usually	ADV
ap-7678	191	40	the	the	DET
ap-7678	191	41	only	only	ADJ
ap-7678	191	42	state	state	NOUN
ap-7678	191	43	that	that	PRON
ap-7678	191	44	can	can	AUX
ap-7678	191	45	be	be	AUX
ap-7678	191	46	determined	determine	VERB
ap-7678	191	47	.	.	PUNCT
ap-7678	192	1	using	use	VERB
ap-7678	192	2	the	the	DET
ap-7678	192	3	ground	ground	NOUN
ap-7678	192	4	state	state	NOUN
ap-7678	192	5	as	as	ADP
ap-7678	192	6	a	a	DET
ap-7678	192	7	weight	weight	NOUN
ap-7678	192	8	usually	usually	ADV
ap-7678	192	9	yields	yield	VERB
ap-7678	192	10	the	the	DET
ap-7678	192	11	fastest	fast	ADJ
ap-7678	192	12	convergence	convergence	NOUN
ap-7678	192	13	.	.	PUNCT
ap-7678	193	1	6	6	X
ap-7678	193	2	.	.	X
ap-7678	194	1	the	the	DET
ap-7678	194	2	oppq	oppq	NOUN
ap-7678	194	3	-	-	PUNCT
ap-7678	194	4	approximation	approximation	NOUN
ap-7678	194	5	method	method	NOUN
ap-7678	194	6	(	(	PUNCT
ap-7678	194	7	oppq	oppq	PROPN
ap-7678	194	8	-	-	PUNCT
ap-7678	194	9	am	am	NOUN
ap-7678	194	10	)	)	PUNCT
ap-7678	194	11	from	from	ADP
ap-7678	194	12	eq	eq	ADP
ap-7678	194	13	.	.	PUNCT
ap-7678	194	14	(	(	PUNCT
ap-7678	194	15	21	21	NUM
ap-7678	194	16	)	)	PUNCT
ap-7678	194	17	and	and	CCONJ
ap-7678	194	18	the	the	DET
ap-7678	194	19	oppq	oppq	ADJ
ap-7678	194	20	quantization	quantization	NOUN
ap-7678	194	21	condition	condition	NOUN
ap-7678	194	22	in	in	ADP
ap-7678	194	23	eq	eq	ADP
ap-7678	194	24	.	.	PUNCT
ap-7678	195	1	(	(	PUNCT
ap-7678	195	2	22	22	NUM
ap-7678	195	3	)	)	PUNCT
ap-7678	195	4	it	it	PRON
ap-7678	195	5	follows	follow	VERB
ap-7678	195	6	that	that	SCONJ
ap-7678	195	7	the	the	DET
ap-7678	195	8	physical	physical	ADJ
ap-7678	195	9	energy	energy	NOUN
ap-7678	195	10	and	and	CCONJ
ap-7678	195	11	missing	missing	ADJ
ap-7678	195	12	moment	moment	NOUN
ap-7678	195	13	values	value	NOUN
ap-7678	195	14	must	must	AUX
ap-7678	195	15	satisfy	satisfy	VERB
ap-7678	195	16	lim→∞cn(ephys	lim→∞cn(ephy	NOUN
ap-7678	195	17	,	,	PUNCT
ap-7678	195	18	−→µ	−→µ	NOUN
ap-7678	195	19	phys	phy	NOUN
ap-7678	195	20	)	)	PUNCT
ap-7678	195	21	=	=	SYM
ap-7678	196	1	0	0	X
ap-7678	196	2	.	.	PUNCT
ap-7678	197	1	(	(	PUNCT
ap-7678	197	2	26	26	NUM
ap-7678	197	3	)	)	PUNCT
ap-7678	197	4	since	since	SCONJ
ap-7678	197	5	cn(e	cn(e	ADJ
ap-7678	197	6	,	,	PUNCT
ap-7678	197	7	−→µ	−→µ	NOUN
ap-7678	197	8	)	)	PUNCT
ap-7678	197	9	=	=	PUNCT
ap-7678	197	10	−→	−→	NOUN
ap-7678	197	11	λ	λ	X
ap-7678	197	12	(	(	PUNCT
ap-7678	197	13	n	n	CCONJ
ap-7678	197	14	)	)	PUNCT
ap-7678	197	15	(	(	PUNCT
ap-7678	197	16	e	e	NOUN
ap-7678	197	17	)	)	PUNCT
ap-7678	197	18	·	·	PUNCT
ap-7678	197	19	−→µ	−→µ	NOUN
ap-7678	197	20	,	,	PUNCT
ap-7678	197	21	from	from	ADP
ap-7678	197	22	eq	eq	ADP
ap-7678	197	23	.	.	PUNCT
ap-7678	198	1	(	(	PUNCT
ap-7678	198	2	17	17	NUM
ap-7678	198	3	)	)	PUNCT
ap-7678	198	4	,	,	PUNCT
ap-7678	198	5	the	the	DET
ap-7678	198	6	1	1	NUM
ap-7678	198	7	+	+	CCONJ
ap-7678	198	8	ms	ms	ADJ
ap-7678	198	9	linear	linear	ADJ
ap-7678	198	10	equations	equation	NOUN
ap-7678	198	11	ms∑	ms∑	VERB
ap-7678	198	12	ℓ2=0	ℓ2=0	PROPN
ap-7678	198	13	λ(n−ℓ1	λ(n−ℓ1	NOUN
ap-7678	198	14	)	)	PUNCT
ap-7678	198	15	ℓ2	ℓ2	NOUN
ap-7678	198	16	(	(	PUNCT
ap-7678	198	17	e	e	NOUN
ap-7678	198	18	)	)	PUNCT
ap-7678	198	19	µℓ2	µℓ2	PROPN
ap-7678	198	20	=	=	SYM
ap-7678	198	21	0	0	NUM
ap-7678	198	22	,	,	PUNCT
ap-7678	198	23	(	(	PUNCT
ap-7678	198	24	27	27	NUM
ap-7678	198	25	)	)	PUNCT
ap-7678	198	26	0	0	NUM
ap-7678	198	27	≤	≤	NOUN
ap-7678	198	28	ℓ1	ℓ1	NOUN
ap-7678	198	29	≤	≤	PUNCT
ap-7678	198	30	ms	ms	NOUN
ap-7678	198	31	,	,	PUNCT
ap-7678	198	32	can	can	AUX
ap-7678	198	33	be	be	AUX
ap-7678	198	34	used	use	VERB
ap-7678	198	35	to	to	PART
ap-7678	198	36	approximate	approximate	VERB
ap-7678	198	37	the	the	DET
ap-7678	198	38	physical	physical	ADJ
ap-7678	198	39	energies	energy	NOUN
ap-7678	198	40	through	through	ADP
ap-7678	198	41	the	the	DET
ap-7678	198	42	determinantal	determinantal	ADJ
ap-7678	198	43	secular	secular	ADJ
ap-7678	198	44	equation	equation	NOUN
ap-7678	198	45	det	det	NOUN
ap-7678	198	46	(	(	PUNCT
ap-7678	198	47	λ(n−ℓ1	λ(n−ℓ1	NOUN
ap-7678	198	48	)	)	PUNCT
ap-7678	198	49	ℓ2	ℓ2	NOUN
ap-7678	198	50	(	(	PUNCT
ap-7678	198	51	e	e	NOUN
ap-7678	198	52	)	)	PUNCT
ap-7678	198	53	)	)	PUNCT
ap-7678	199	1	=	=	PUNCT
ap-7678	199	2	0	0	X
ap-7678	199	3	.	.	PUNCT
ap-7678	200	1	(	(	PUNCT
ap-7678	200	2	28	28	NUM
ap-7678	200	3	)	)	PUNCT
ap-7678	200	4	this	this	PRON
ap-7678	200	5	defines	define	VERB
ap-7678	200	6	the	the	DET
ap-7678	200	7	oppq	oppq	NOUN
ap-7678	200	8	-	-	PUNCT
ap-7678	200	9	approximation	approximation	NOUN
ap-7678	200	10	method	method	NOUN
ap-7678	200	11	(	(	PUNCT
ap-7678	200	12	oppq	oppq	PROPN
ap-7678	200	13	-	-	PUNCT
ap-7678	200	14	am	am	NOUN
ap-7678	200	15	)	)	PUNCT
ap-7678	200	16	.	.	PUNCT
ap-7678	201	1	as	as	SCONJ
ap-7678	201	2	indicated	indicate	VERB
ap-7678	201	3	earlier	early	ADV
ap-7678	201	4	,	,	PUNCT
ap-7678	201	5	handy	handy	ADJ
ap-7678	201	6	and	and	CCONJ
ap-7678	201	7	vrinceanu	vrinceanu	NOUN
ap-7678	202	1	[	[	X
ap-7678	202	2	10	10	NUM
ap-7678	202	3	,	,	PUNCT
ap-7678	202	4	11	11	NUM
ap-7678	202	5	]	]	PUNCT
ap-7678	202	6	noted	note	VERB
ap-7678	202	7	significant	significant	ADJ
ap-7678	202	8	improvement	improvement	NOUN
ap-7678	202	9	in	in	ADP
ap-7678	202	10	the	the	DET
ap-7678	202	11	convergence	convergence	NOUN
ap-7678	202	12	rates	rate	NOUN
ap-7678	202	13	when	when	SCONJ
ap-7678	202	14	the	the	DET
ap-7678	202	15	weight	weight	NOUN
ap-7678	202	16	,	,	PUNCT
ap-7678	202	17	r	r	NOUN
ap-7678	202	18	,	,	PUNCT
ap-7678	202	19	mimics	mimic	VERB
ap-7678	202	20	the	the	DET
ap-7678	202	21	asymptotic	asymptotic	ADJ
ap-7678	202	22	form	form	NOUN
ap-7678	202	23	of	of	ADP
ap-7678	202	24	the	the	DET
ap-7678	202	25	desired	desire	VERB
ap-7678	202	26	physical	physical	ADJ
ap-7678	202	27	states	state	NOUN
ap-7678	202	28	.	.	PUNCT
ap-7678	203	1	the	the	DET
ap-7678	203	2	oppq	oppq	ADJ
ap-7678	203	3	representation	representation	NOUN
ap-7678	203	4	is	be	AUX
ap-7678	203	5	very	very	ADV
ap-7678	203	6	robust	robust	ADJ
ap-7678	203	7	,	,	PUNCT
ap-7678	203	8	and	and	CCONJ
ap-7678	203	9	no	no	DET
ap-7678	203	10	convergence	convergence	NOUN
ap-7678	203	11	irregularities	irregularity	NOUN
ap-7678	203	12	emerge	emerge	VERB
ap-7678	203	13	so	so	ADV
ap-7678	203	14	long	long	ADV
ap-7678	203	15	as	as	SCONJ
ap-7678	203	16	the	the	DET
ap-7678	203	17	weight	weight	NOUN
ap-7678	203	18	does	do	AUX
ap-7678	203	19	not	not	PART
ap-7678	203	20	decrease	decrease	VERB
ap-7678	203	21	much	much	ADV
ap-7678	203	22	faster	fast	ADV
ap-7678	203	23	than	than	ADP
ap-7678	203	24	the	the	DET
ap-7678	203	25	asymptotic	asymptotic	ADJ
ap-7678	203	26	form	form	NOUN
ap-7678	203	27	of	of	ADP
ap-7678	203	28	the	the	DET
ap-7678	203	29	physical	physical	ADJ
ap-7678	203	30	states	state	NOUN
ap-7678	203	31	.	.	PUNCT
ap-7678	204	1	provided	provide	VERB
ap-7678	204	2	this	this	PRON
ap-7678	204	3	is	be	AUX
ap-7678	204	4	satisfied	satisfied	ADJ
ap-7678	204	5	,	,	PUNCT
ap-7678	204	6	the	the	DET
ap-7678	204	7	oppq	oppq	NOUN
ap-7678	204	8	-	-	PUNCT
ap-7678	204	9	am	am	NOUN
ap-7678	204	10	formalism	formalism	NOUN
ap-7678	204	11	will	will	AUX
ap-7678	204	12	always	always	ADV
ap-7678	204	13	converge	converge	VERB
ap-7678	204	14	to	to	ADP
ap-7678	204	15	the	the	DET
ap-7678	204	16	true	true	ADJ
ap-7678	204	17	physical	physical	ADJ
ap-7678	204	18	energies	energy	NOUN
ap-7678	204	19	;	;	PUNCT
ap-7678	204	20	however	however	ADV
ap-7678	204	21	,	,	PUNCT
ap-7678	204	22	the	the	DET
ap-7678	204	23	rate	rate	NOUN
ap-7678	204	24	of	of	ADP
ap-7678	204	25	convergence	convergence	NOUN
ap-7678	204	26	depends	depend	VERB
ap-7678	204	27	on	on	ADP
ap-7678	204	28	the	the	DET
ap-7678	204	29	asymptotic	asymptotic	ADJ
ap-7678	204	30	properties	property	NOUN
ap-7678	204	31	of	of	ADP
ap-7678	204	32	the	the	DET
ap-7678	204	33	chosen	choose	VERB
ap-7678	204	34	weight	weight	NOUN
ap-7678	204	35	.	.	PUNCT
ap-7678	205	1	despite	despite	SCONJ
ap-7678	205	2	these	these	DET
ap-7678	205	3	impressive	impressive	ADJ
ap-7678	205	4	results	result	NOUN
ap-7678	205	5	,	,	PUNCT
ap-7678	205	6	there	there	PRON
ap-7678	205	7	is	be	VERB
ap-7678	205	8	no	no	DET
ap-7678	205	9	guarantee	guarantee	NOUN
ap-7678	205	10	that	that	SCONJ
ap-7678	205	11	,	,	PUNCT
ap-7678	205	12	for	for	ADP
ap-7678	205	13	hermitian	hermitian	ADJ
ap-7678	205	14	systems	system	NOUN
ap-7678	205	15	,	,	PUNCT
ap-7678	205	16	the	the	DET
ap-7678	205	17	energies	energy	NOUN
ap-7678	205	18	generated	generate	VERB
ap-7678	205	19	through	through	ADP
ap-7678	205	20	eq	eq	PROPN
ap-7678	205	21	.	.	PUNCT
ap-7678	206	1	(	(	PUNCT
ap-7678	206	2	28	28	NUM
ap-7678	206	3	)	)	PUNCT
ap-7678	206	4	will	will	AUX
ap-7678	206	5	be	be	AUX
ap-7678	206	6	real	real	ADJ
ap-7678	206	7	.	.	PUNCT
ap-7678	207	1	spurious	spurious	ADJ
ap-7678	207	2	,	,	PUNCT
ap-7678	207	3	small	small	ADJ
ap-7678	207	4	imaginary	imaginary	ADJ
ap-7678	207	5	parts	part	NOUN
ap-7678	207	6	may	may	AUX
ap-7678	207	7	be	be	AUX
ap-7678	207	8	produced	produce	VERB
ap-7678	207	9	from	from	ADP
ap-7678	207	10	eq	eq	PROPN
ap-7678	207	11	.	.	PUNCT
ap-7678	208	1	(	(	PUNCT
ap-7678	208	2	28	28	NUM
ap-7678	208	3	)	)	PUNCT
ap-7678	208	4	,	,	PUNCT
ap-7678	208	5	that	that	PRON
ap-7678	208	6	vanish	vanish	VERB
ap-7678	208	7	in	in	ADP
ap-7678	208	8	the	the	DET
ap-7678	208	9	n	n	NOUN
ap-7678	208	10	→	→	SYM
ap-7678	208	11	∞	∞	NUM
ap-7678	208	12	limit	limit	NOUN
ap-7678	208	13	.	.	PUNCT
ap-7678	209	1	by	by	ADP
ap-7678	209	2	way	way	NOUN
ap-7678	209	3	of	of	ADP
ap-7678	209	4	contrast	contrast	NOUN
ap-7678	209	5	,	,	PUNCT
ap-7678	209	6	the	the	DET
ap-7678	209	7	oppq	oppq	NOUN
ap-7678	209	8	-	-	PUNCT
ap-7678	209	9	bounding	bound	VERB
ap-7678	209	10	method	method	NOUN
ap-7678	209	11	to	to	PART
ap-7678	209	12	be	be	AUX
ap-7678	209	13	described	describe	VERB
ap-7678	209	14	below	below	ADV
ap-7678	209	15	,	,	PUNCT
ap-7678	209	16	will	will	AUX
ap-7678	209	17	always	always	ADV
ap-7678	209	18	generate	generate	VERB
ap-7678	209	19	real	real	ADJ
ap-7678	209	20	energies	energy	NOUN
ap-7678	209	21	for	for	ADP
ap-7678	209	22	hermitian	hermitian	ADJ
ap-7678	209	23	systems	system	NOUN
ap-7678	209	24	.	.	PUNCT
ap-7678	210	1	in	in	ADP
ap-7678	210	2	addition	addition	NOUN
ap-7678	210	3	,	,	PUNCT
ap-7678	210	4	as	as	SCONJ
ap-7678	210	5	its	its	PRON
ap-7678	210	6	name	name	NOUN
ap-7678	210	7	suggests	suggest	VERB
ap-7678	210	8	,	,	PUNCT
ap-7678	210	9	converging	converge	VERB
ap-7678	210	10	bounds	bound	NOUN
ap-7678	210	11	to	to	ADP
ap-7678	210	12	the	the	DET
ap-7678	210	13	discrete	discrete	ADJ
ap-7678	210	14	state	state	NOUN
ap-7678	210	15	energies	energy	NOUN
ap-7678	210	16	can	can	AUX
ap-7678	210	17	be	be	AUX
ap-7678	210	18	generated	generate	VERB
ap-7678	210	19	.	.	PUNCT
ap-7678	211	1	we	we	PRON
ap-7678	211	2	make	make	VERB
ap-7678	211	3	the	the	DET
ap-7678	211	4	last	last	ADJ
ap-7678	211	5	observation	observation	NOUN
ap-7678	211	6	more	more	ADV
ap-7678	211	7	explicit	explicit	ADJ
ap-7678	211	8	.	.	PUNCT
ap-7678	212	1	as	as	SCONJ
ap-7678	212	2	will	will	AUX
ap-7678	212	3	be	be	AUX
ap-7678	212	4	shown	show	VERB
ap-7678	212	5	below	below	ADP
ap-7678	212	6	,	,	PUNCT
ap-7678	212	7	oppq	oppq	PROPN
ap-7678	212	8	-	-	PUNCT
ap-7678	212	9	bm	bm	NOUN
ap-7678	212	10	generates	generate	NOUN
ap-7678	212	11	,	,	PUNCT
ap-7678	212	12	to	to	ADP
ap-7678	212	13	each	each	DET
ap-7678	212	14	order	order	NOUN
ap-7678	212	15	,	,	PUNCT
ap-7678	212	16	an	an	DET
ap-7678	212	17	energy	energy	NOUN
ap-7678	212	18	dependent	dependent	ADJ
ap-7678	212	19	function	function	NOUN
ap-7678	212	20	,	,	PUNCT
ap-7678	212	21	ln	ln	X
ap-7678	212	22	(	(	PUNCT
ap-7678	212	23	e	e	NOUN
ap-7678	212	24	)	)	PUNCT
ap-7678	212	25	.	.	PUNCT
ap-7678	213	1	the	the	DET
ap-7678	213	2	local	local	ADJ
ap-7678	213	3	minima	minima	NOUN
ap-7678	213	4	,	,	PUNCT
ap-7678	213	5	∂eln	∂eln	PUNCT
ap-7678	213	6	(	(	PUNCT
ap-7678	213	7	e	e	NOUN
ap-7678	213	8	)	)	PUNCT
ap-7678	213	9	=	=	SYM
ap-7678	213	10	0	0	PUNCT
ap-7678	213	11	will	will	AUX
ap-7678	213	12	approximate	approximate	VERB
ap-7678	213	13	the	the	DET
ap-7678	213	14	physical	physical	ADJ
ap-7678	213	15	energies	energy	NOUN
ap-7678	213	16	.	.	PUNCT
ap-7678	214	1	these	these	DET
ap-7678	214	2	local	local	ADJ
ap-7678	214	3	minima	minima	NOUN
ap-7678	214	4	can	can	AUX
ap-7678	214	5	,	,	PUNCT
ap-7678	214	6	essentially	essentially	ADV
ap-7678	214	7	,	,	PUNCT
ap-7678	214	8	be	be	AUX
ap-7678	214	9	bounded	bound	VERB
ap-7678	214	10	,	,	PUNCT
ap-7678	214	11	through	through	ADP
ap-7678	214	12	converging	converge	VERB
ap-7678	214	13	lower	low	ADJ
ap-7678	214	14	and	and	CCONJ
ap-7678	214	15	upper	upper	ADJ
ap-7678	214	16	bounds	bound	NOUN
ap-7678	214	17	.	.	PUNCT
ap-7678	215	1	we	we	PRON
ap-7678	215	2	refer	refer	VERB
ap-7678	215	3	to	to	ADP
ap-7678	215	4	the	the	DET
ap-7678	215	5	local	local	ADJ
ap-7678	215	6	minima	minima	NOUN
ap-7678	215	7	in	in	ADP
ap-7678	215	8	ln	ln	PROPN
ap-7678	215	9	(	(	PUNCT
ap-7678	215	10	e	e	NOUN
ap-7678	215	11	)	)	PUNCT
ap-7678	215	12	as	as	SCONJ
ap-7678	215	13	the	the	DET
ap-7678	215	14	oppqbm	oppqbm	NOUN
ap-7678	215	15	estimates	estimate	VERB
ap-7678	215	16	,	,	PUNCT
ap-7678	215	17	in	in	ADP
ap-7678	215	18	order	order	NOUN
ap-7678	215	19	to	to	PART
ap-7678	215	20	distinguish	distinguish	VERB
ap-7678	215	21	them	they	PRON
ap-7678	215	22	from	from	ADP
ap-7678	215	23	the	the	DET
ap-7678	215	24	oppq	oppq	NOUN
ap-7678	215	25	-	-	PUNCT
ap-7678	215	26	am	am	NOUN
ap-7678	215	27	estimates	estimate	NOUN
ap-7678	215	28	from	from	ADP
ap-7678	215	29	eq	eq	PROPN
ap-7678	215	30	.	.	PUNCT
ap-7678	216	1	(	(	PUNCT
ap-7678	216	2	28	28	NUM
ap-7678	216	3	)	)	PUNCT
ap-7678	216	4	and	and	CCONJ
ap-7678	216	5	the	the	DET
ap-7678	216	6	oppqbm	oppqbm	NOUN
ap-7678	216	7	generated	generate	VERB
ap-7678	216	8	bounds	bound	NOUN
ap-7678	216	9	.	.	PUNCT
ap-7678	217	1	7	7	X
ap-7678	217	2	.	.	X
ap-7678	217	3	the	the	DET
ap-7678	217	4	oppq	oppq	NOUN
ap-7678	217	5	-	-	PUNCT
ap-7678	217	6	bounding	bound	VERB
ap-7678	217	7	method	method	NOUN
ap-7678	217	8	we	we	PRON
ap-7678	217	9	outline	outline	VERB
ap-7678	217	10	the	the	DET
ap-7678	217	11	structure	structure	NOUN
ap-7678	217	12	of	of	ADP
ap-7678	217	13	oppq	oppq	PROPN
ap-7678	217	14	-	-	PUNCT
ap-7678	217	15	bm	bm	PROPN
ap-7678	217	16	for	for	ADP
ap-7678	217	17	one	one	NUM
ap-7678	217	18	space	space	NOUN
ap-7678	217	19	dimension	dimension	NOUN
ap-7678	217	20	problems	problem	NOUN
ap-7678	217	21	.	.	PUNCT
ap-7678	218	1	the	the	DET
ap-7678	218	2	major	major	ADJ
ap-7678	218	3	difference	difference	NOUN
ap-7678	218	4	between	between	ADP
ap-7678	218	5	one	one	NUM
ap-7678	218	6	dimensional	dimensional	ADJ
ap-7678	218	7	and	and	CCONJ
ap-7678	218	8	multidimensional	multidimensional	ADJ
ap-7678	218	9	mer	mer	NOUN
ap-7678	218	10	type	type	NOUN
ap-7678	218	11	systems	system	NOUN
ap-7678	218	12	is	be	AUX
ap-7678	218	13	that	that	SCONJ
ap-7678	218	14	one	one	NUM
ap-7678	218	15	dimensional	dimensional	ADJ
ap-7678	218	16	problems	problem	NOUN
ap-7678	218	17	have	have	VERB
ap-7678	218	18	a	a	DET
ap-7678	218	19	fixed	fix	VERB
ap-7678	218	20	number	number	NOUN
ap-7678	218	21	of	of	ADP
ap-7678	218	22	missing	miss	VERB
ap-7678	218	23	moments	moment	NOUN
ap-7678	218	24	:	:	PUNCT
ap-7678	218	25	ms	ms	PROPN
ap-7678	218	26	<	<	X
ap-7678	218	27	∞.	∞.	PROPN
ap-7678	218	28	multidimensional	multidimensional	ADJ
ap-7678	218	29	problems	problem	NOUN
ap-7678	218	30	involve	involve	VERB
ap-7678	218	31	an	an	DET
ap-7678	218	32	infinite	infinite	ADJ
ap-7678	218	33	hierarchy	hierarchy	NOUN
ap-7678	218	34	of	of	ADP
ap-7678	218	35	missing	miss	VERB
ap-7678	218	36	moment	moment	NOUN
ap-7678	218	37	subspaces	subspace	NOUN
ap-7678	218	38	of	of	ADP
ap-7678	218	39	increasing	increase	VERB
ap-7678	218	40	dimension	dimension	NOUN
ap-7678	218	41	.	.	PUNCT
ap-7678	219	1	that	that	PRON
ap-7678	219	2	is	be	AUX
ap-7678	219	3	,	,	PUNCT
ap-7678	219	4	ms	ms	PROPN
ap-7678	219	5	→	→	SYM
ap-7678	219	6	∞.	∞.	PROPN
ap-7678	219	7	we	we	PRON
ap-7678	219	8	develop	develop	VERB
ap-7678	219	9	the	the	DET
ap-7678	219	10	1space	1space	NUM
ap-7678	219	11	dimension	dimension	NOUN
ap-7678	219	12	oppq	oppq	ADJ
ap-7678	219	13	formalism	formalism	NOUN
ap-7678	219	14	in	in	ADP
ap-7678	219	15	a	a	DET
ap-7678	219	16	manner	manner	NOUN
ap-7678	219	17	that	that	PRON
ap-7678	219	18	extends	extend	VERB
ap-7678	219	19	to	to	ADP
ap-7678	219	20	multidimensions	multidimension	NOUN
ap-7678	219	21	.	.	PUNCT
ap-7678	220	1	how	how	SCONJ
ap-7678	220	2	the	the	DET
ap-7678	220	3	normalization	normalization	NOUN
ap-7678	220	4	prescription	prescription	NOUN
ap-7678	220	5	is	be	AUX
ap-7678	220	6	chosen	choose	VERB
ap-7678	220	7	,	,	PUNCT
ap-7678	220	8	plays	play	VERB
ap-7678	220	9	an	an	DET
ap-7678	220	10	important	important	ADJ
ap-7678	220	11	role	role	NOUN
ap-7678	220	12	in	in	ADP
ap-7678	220	13	the	the	DET
ap-7678	220	14	formalism	formalism	NOUN
ap-7678	220	15	[	[	X
ap-7678	220	16	9	9	NUM
ap-7678	220	17	]	]	PUNCT
ap-7678	220	18	.	.	PUNCT
ap-7678	221	1	67	67	NUM
ap-7678	221	2	carlos	carlos	PROPN
ap-7678	221	3	r.	r.	PROPN
ap-7678	221	4	handy	handy	PROPN
ap-7678	221	5	acta	acta	PROPN
ap-7678	221	6	polytechnica	polytechnica	PROPN
ap-7678	221	7	5	5	NUM
ap-7678	221	8	10	10	NUM
ap-7678	221	9	15	15	NUM
ap-7678	221	10	20	20	NUM
ap-7678	221	11	e	e	NOUN
ap-7678	221	12	35	35	NUM
ap-7678	221	13	log	log	NOUN
ap-7678	221	14	10	10	NUM
ap-7678	221	15	(	(	PUNCT
ap-7678	221	16	�	�	NOUN
ap-7678	221	17	n(e	n(e	NOUN
ap-7678	221	18	)	)	PUNCT
ap-7678	221	19	)	)	PUNCT
ap-7678	221	20	figure	figure	NOUN
ap-7678	221	21	1	1	NUM
ap-7678	221	22	.	.	X
ap-7678	221	23	log10(λn	log10(λn	PROPN
ap-7678	221	24	(	(	PUNCT
ap-7678	221	25	e	e	NOUN
ap-7678	221	26	)	)	PUNCT
ap-7678	221	27	)	)	PUNCT
ap-7678	221	28	for	for	ADP
ap-7678	221	29	sextic	sextic	ADJ
ap-7678	221	30	anharmonic	anharmonic	ADJ
ap-7678	221	31	oscillator	oscillator	NOUN
ap-7678	221	32	system	system	NOUN
ap-7678	221	33	in	in	ADP
ap-7678	221	34	eq	eq	ADP
ap-7678	221	35	.	.	PUNCT
ap-7678	222	1	(	(	PUNCT
ap-7678	222	2	47	47	NUM
ap-7678	222	3	)	)	PUNCT
ap-7678	222	4	;	;	PUNCT
ap-7678	222	5	n	n	PROPN
ap-7678	222	6	=	=	NUM
ap-7678	222	7	20	20	NUM
ap-7678	222	8	,	,	PUNCT
ap-7678	222	9	40	40	NUM
ap-7678	222	10	,	,	PUNCT
ap-7678	222	11	100	100	NUM
ap-7678	222	12	,	,	PUNCT
ap-7678	222	13	120	120	NUM
ap-7678	222	14	.	.	PUNCT
ap-7678	223	1	the	the	DET
ap-7678	223	2	oppq	oppq	ADJ
ap-7678	223	3	quantization	quantization	NOUN
ap-7678	223	4	condition	condition	NOUN
ap-7678	223	5	in	in	ADP
ap-7678	223	6	eq	eq	ADP
ap-7678	223	7	.	.	PUNCT
ap-7678	224	1	(	(	PUNCT
ap-7678	224	2	22	22	NUM
ap-7678	224	3	)	)	PUNCT
ap-7678	224	4	is	be	AUX
ap-7678	224	5	dependent	dependent	ADJ
ap-7678	224	6	on	on	ADP
ap-7678	224	7	the	the	DET
ap-7678	224	8	positive	positive	ADJ
ap-7678	224	9	,	,	PUNCT
ap-7678	224	10	(	(	PUNCT
ap-7678	224	11	essentially	essentially	ADV
ap-7678	224	12	)	)	PUNCT
ap-7678	224	13	increasing	increase	VERB
ap-7678	224	14	,	,	PUNCT
ap-7678	224	15	sequence	sequence	NOUN
ap-7678	224	16	,	,	PUNCT
ap-7678	224	17	defined	define	VERB
ap-7678	224	18	by	by	ADP
ap-7678	224	19	the	the	DET
ap-7678	224	20	partial	partial	ADJ
ap-7678	224	21	sums	sum	NOUN
ap-7678	224	22	:	:	PUNCT
ap-7678	224	23	i[ψ	i[ψ	NOUN
ap-7678	224	24	,	,	PUNCT
ap-7678	224	25	r	r	NOUN
ap-7678	224	26	]	]	X
ap-7678	224	27	=	=	PUNCT
ap-7678	224	28	limn→∞sn	limn→∞sn	NOUN
ap-7678	224	29	(	(	PUNCT
ap-7678	224	30	e	e	NOUN
ap-7678	224	31	,	,	PUNCT
ap-7678	224	32	−→µ	−→µ	NOUN
ap-7678	224	33	)	)	PUNCT
ap-7678	224	34	,	,	PUNCT
ap-7678	224	35	(	(	PUNCT
ap-7678	224	36	29	29	NUM
ap-7678	224	37	)	)	PUNCT
ap-7678	224	38	where	where	SCONJ
ap-7678	224	39	sn	sn	PROPN
ap-7678	224	40	(	(	PUNCT
ap-7678	224	41	e	e	NOUN
ap-7678	224	42	,	,	PUNCT
ap-7678	224	43	−→µ	−→µ	NOUN
ap-7678	224	44	)	)	PUNCT
ap-7678	224	45	=	=	SYM
ap-7678	224	46	n∑	n∑	X
ap-7678	224	47	j=0	j=0	PROPN
ap-7678	224	48	|cj(e	|cj(e	PROPN
ap-7678	224	49	,	,	PUNCT
ap-7678	224	50	−→µ	−→µ	NOUN
ap-7678	224	51	)	)	PUNCT
ap-7678	224	52	|2	|2	NUM
ap-7678	224	53	,	,	PUNCT
ap-7678	224	54	(	(	PUNCT
ap-7678	224	55	30	30	NUM
ap-7678	224	56	)	)	PUNCT
ap-7678	224	57	=	=	SYM
ap-7678	224	58	⟨−→µ	⟨−→µ	NOUN
ap-7678	224	59	|pn	|pn	X
ap-7678	224	60	(	(	PUNCT
ap-7678	224	61	e)|−→µ	e)|−→µ	NOUN
ap-7678	224	62	⟩	⟩	NOUN
ap-7678	224	63	,	,	PUNCT
ap-7678	224	64	(	(	PUNCT
ap-7678	224	65	31	31	NUM
ap-7678	224	66	)	)	PUNCT
ap-7678	224	67	where	where	SCONJ
ap-7678	224	68	pn	pn	PROPN
ap-7678	224	69	(	(	PUNCT
ap-7678	224	70	e	e	NOUN
ap-7678	224	71	)	)	PUNCT
ap-7678	224	72	is	be	AUX
ap-7678	224	73	an	an	DET
ap-7678	224	74	energy	energy	NOUN
ap-7678	224	75	dependent	dependent	ADJ
ap-7678	224	76	,	,	PUNCT
ap-7678	224	77	positive	positive	ADJ
ap-7678	224	78	matrix	matrix	NOUN
ap-7678	224	79	(	(	PUNCT
ap-7678	224	80	if	if	SCONJ
ap-7678	224	81	n	n	PRON
ap-7678	224	82	≥	≥	X
ap-7678	224	83	ms	ms	NOUN
ap-7678	224	84	)	)	PUNCT
ap-7678	224	85	of	of	ADP
ap-7678	224	86	dimension	dimension	NOUN
ap-7678	224	87	(	(	PUNCT
ap-7678	224	88	1	1	NUM
ap-7678	224	89	+	+	CCONJ
ap-7678	224	90	ms	ms	PROPN
ap-7678	224	91	)	)	PUNCT
ap-7678	224	92	×	×	NOUN
ap-7678	224	93	(	(	PUNCT
ap-7678	224	94	1	1	NUM
ap-7678	224	95	+	+	CCONJ
ap-7678	224	96	ms	ms	PROPN
ap-7678	224	97	):	):	PUNCT
ap-7678	224	98	pn	pn	PROPN
ap-7678	224	99	(	(	PUNCT
ap-7678	224	100	e	e	NOUN
ap-7678	224	101	)	)	PUNCT
ap-7678	225	1	=	=	SYM
ap-7678	225	2	n∑	n∑	X
ap-7678	225	3	j=0	j=0	PROPN
ap-7678	225	4	(	(	PUNCT
ap-7678	225	5	−→	−→	NOUN
ap-7678	225	6	λ	λ	PROPN
ap-7678	225	7	(	(	PUNCT
ap-7678	225	8	j	j	NOUN
ap-7678	225	9	)	)	PUNCT
ap-7678	225	10	(	(	PUNCT
ap-7678	225	11	e	e	NOUN
ap-7678	225	12	)	)	PUNCT
ap-7678	225	13	)	)	PUNCT
ap-7678	225	14	∗−→	∗−→	PROPN
ap-7678	225	15	λ	λ	PROPN
ap-7678	225	16	(	(	PUNCT
ap-7678	225	17	j	j	NOUN
ap-7678	225	18	)	)	PUNCT
ap-7678	225	19	(	(	PUNCT
ap-7678	225	20	e	e	NOUN
ap-7678	225	21	)	)	PUNCT
ap-7678	225	22	,	,	PUNCT
ap-7678	225	23	(	(	PUNCT
ap-7678	225	24	32	32	NUM
ap-7678	225	25	)	)	PUNCT
ap-7678	225	26	involving	involve	VERB
ap-7678	225	27	the	the	DET
ap-7678	225	28	sum	sum	NOUN
ap-7678	225	29	over	over	ADP
ap-7678	225	30	dyad	dyad	NOUN
ap-7678	225	31	matrix	matrix	NOUN
ap-7678	225	32	expressions	expression	NOUN
ap-7678	225	33	.	.	PUNCT
ap-7678	226	1	for	for	ADP
ap-7678	226	2	non	non	ADJ
ap-7678	226	3	-	-	ADJ
ap-7678	226	4	hermitian	hermitian	ADJ
ap-7678	226	5	systems	system	NOUN
ap-7678	226	6	,	,	PUNCT
ap-7678	226	7	the	the	DET
ap-7678	226	8	“	"	PUNCT
ap-7678	226	9	bra	bra	NOUN
ap-7678	226	10	”	"	PUNCT
ap-7678	226	11	missing	miss	VERB
ap-7678	226	12	moment	moment	NOUN
ap-7678	226	13	vector	vector	NOUN
ap-7678	226	14	in	in	ADP
ap-7678	226	15	eq	eq	ADP
ap-7678	226	16	.	.	PUNCT
ap-7678	227	1	(	(	PUNCT
ap-7678	227	2	31	31	NUM
ap-7678	227	3	)	)	PUNCT
ap-7678	227	4	requires	require	VERB
ap-7678	227	5	the	the	DET
ap-7678	227	6	complex	complex	ADJ
ap-7678	227	7	conjugate	conjugate	ADJ
ap-7678	227	8	expression	expression	NOUN
ap-7678	227	9	for	for	ADP
ap-7678	227	10	positive	positive	ADJ
ap-7678	227	11	norms	norm	NOUN
ap-7678	227	12	on	on	ADP
ap-7678	227	13	complex	complex	ADJ
ap-7678	227	14	vector	vector	NOUN
ap-7678	227	15	spaces	space	NOUN
ap-7678	227	16	.	.	PUNCT
ap-7678	228	1	it	it	PRON
ap-7678	228	2	trivially	trivially	ADV
ap-7678	228	3	follows	follow	VERB
ap-7678	228	4	,	,	PUNCT
ap-7678	228	5	by	by	ADP
ap-7678	228	6	definition	definition	NOUN
ap-7678	228	7	,	,	PUNCT
ap-7678	228	8	that	that	SCONJ
ap-7678	228	9	0	0	PUNCT
ap-7678	228	10	<	<	X
ap-7678	228	11	sn	sn	PROPN
ap-7678	228	12	(	(	PUNCT
ap-7678	228	13	e	e	NOUN
ap-7678	228	14	,	,	PUNCT
ap-7678	228	15	−→µ	−→µ	NOUN
ap-7678	228	16	)	)	PUNCT
ap-7678	228	17	<	<	X
ap-7678	228	18	sn+1(e	sn+1(e	INTJ
ap-7678	228	19	,	,	PUNCT
ap-7678	228	20	−→µ	−→µ	NOUN
ap-7678	228	21	)	)	PUNCT
ap-7678	228	22	<	<	X
ap-7678	228	23	.	.	PUNCT
ap-7678	228	24	.	.	PUNCT
ap-7678	228	25	.	.	PUNCT
ap-7678	229	1	<	<	X
ap-7678	229	2	i[e	i[e	ADJ
ap-7678	229	3	,	,	PUNCT
ap-7678	229	4	−→µ	−→µ	NOUN
ap-7678	229	5	]	]	PUNCT
ap-7678	229	6	.	.	PUNCT
ap-7678	230	1	(	(	PUNCT
ap-7678	230	2	33	33	NUM
ap-7678	230	3	)	)	PUNCT
ap-7678	230	4	the	the	DET
ap-7678	230	5	oppq	oppq	ADJ
ap-7678	230	6	quantization	quantization	NOUN
ap-7678	230	7	condition	condition	NOUN
ap-7678	230	8	in	in	ADP
ap-7678	230	9	eq	eq	ADP
ap-7678	230	10	.	.	PUNCT
ap-7678	231	1	(	(	PUNCT
ap-7678	231	2	22	22	NUM
ap-7678	231	3	)	)	PUNCT
ap-7678	231	4	tells	tell	VERB
ap-7678	231	5	us	we	PRON
ap-7678	231	6	that	that	SCONJ
ap-7678	231	7	the	the	DET
ap-7678	231	8	physical	physical	ADJ
ap-7678	231	9	energy	energy	NOUN
ap-7678	231	10	and	and	CCONJ
ap-7678	231	11	corresponding	correspond	VERB
ap-7678	231	12	missing	missing	ADJ
ap-7678	231	13	moments	moment	NOUN
ap-7678	231	14	correspond	correspond	VERB
ap-7678	231	15	to	to	ADP
ap-7678	231	16	(	(	PUNCT
ap-7678	231	17	ephys	ephy	NOUN
ap-7678	231	18	,	,	PUNCT
ap-7678	231	19	−→µ	−→µ	NOUN
ap-7678	231	20	phys	phy	NOUN
ap-7678	231	21	)	)	PUNCT
ap-7678	231	22	points	point	NOUN
ap-7678	231	23	within	within	ADP
ap-7678	231	24	the	the	DET
ap-7678	231	25	e	e	NOUN
ap-7678	231	26	×	×	NOUN
ap-7678	231	27	−→µ	−→µ	NOUN
ap-7678	231	28	parameter	parameter	NOUN
ap-7678	231	29	space	space	NOUN
ap-7678	231	30	where	where	SCONJ
ap-7678	231	31	the	the	DET
ap-7678	231	32	functional	functional	ADJ
ap-7678	231	33	i[ψ	i[ψ	NOUN
ap-7678	231	34	,	,	PUNCT
ap-7678	231	35	r	r	X
ap-7678	231	36	]	]	X
ap-7678	231	37	has	have	VERB
ap-7678	231	38	a	a	DET
ap-7678	231	39	local	local	ADJ
ap-7678	231	40	minimum	minimum	NOUN
ap-7678	231	41	.	.	PUNCT
ap-7678	232	1	also	also	ADV
ap-7678	232	2	,	,	PUNCT
ap-7678	232	3	for	for	ADP
ap-7678	232	4	fixed	fix	VERB
ap-7678	232	5	ephys	ephy	NOUN
ap-7678	232	6	,	,	PUNCT
ap-7678	232	7	the	the	DET
ap-7678	232	8	corresponding	corresponding	ADJ
ap-7678	232	9	physical	physical	ADJ
ap-7678	232	10	missing	missing	ADJ
ap-7678	232	11	moment	moment	NOUN
ap-7678	232	12	values	value	NOUN
ap-7678	232	13	are	be	AUX
ap-7678	232	14	those	those	PRON
ap-7678	232	15	corresponding	correspond	VERB
ap-7678	232	16	to	to	ADP
ap-7678	232	17	a	a	DET
ap-7678	232	18	global	global	ADJ
ap-7678	232	19	minimum	minimum	NOUN
ap-7678	232	20	in	in	ADP
ap-7678	232	21	the	the	DET
ap-7678	232	22	missing	missing	ADJ
ap-7678	232	23	moment	moment	NOUN
ap-7678	232	24	space	space	NOUN
ap-7678	232	25	.	.	PUNCT
ap-7678	233	1	therefore	therefore	ADV
ap-7678	233	2	,	,	PUNCT
ap-7678	233	3	to	to	PART
ap-7678	233	4	order	order	VERB
ap-7678	233	5	n	n	PRON
ap-7678	233	6	we	we	PRON
ap-7678	233	7	can	can	AUX
ap-7678	233	8	focus	focus	VERB
ap-7678	233	9	on	on	ADP
ap-7678	233	10	the	the	DET
ap-7678	233	11	global	global	ADJ
ap-7678	233	12	minimum	minimum	NOUN
ap-7678	233	13	within	within	ADP
ap-7678	233	14	the	the	DET
ap-7678	233	15	constrained	constrain	VERB
ap-7678	233	16	(	(	PUNCT
ap-7678	233	17	i.e.	i.e.	X
ap-7678	233	18	normalized	normalize	VERB
ap-7678	233	19	)	)	PUNCT
ap-7678	233	20	missing	miss	VERB
ap-7678	233	21	moment	moment	NOUN
ap-7678	233	22	space	space	NOUN
ap-7678	233	23	:	:	PUNCT
ap-7678	233	24	ln	ln	X
ap-7678	233	25	(	(	PUNCT
ap-7678	233	26	e	e	NOUN
ap-7678	233	27	)	)	PUNCT
ap-7678	233	28	≡	≡	PROPN
ap-7678	233	29	inf−→µ	inf−→µ	PROPN
ap-7678	233	30	{	{	PUNCT
ap-7678	233	31	sn	sn	X
ap-7678	233	32	(	(	PUNCT
ap-7678	233	33	e	e	NOUN
ap-7678	233	34	,	,	PUNCT
ap-7678	233	35	−→µ	−→µ	NOUN
ap-7678	233	36	)	)	PUNCT
ap-7678	233	37	|cnorm(−→µ	|cnorm(−→µ	PROPN
ap-7678	233	38	)	)	PUNCT
ap-7678	234	1	=	=	PUNCT
ap-7678	234	2	1	1	X
ap-7678	234	3	}	}	PUNCT
ap-7678	234	4	,	,	PUNCT
ap-7678	234	5	(	(	PUNCT
ap-7678	234	6	34	34	NUM
ap-7678	234	7	)	)	PUNCT
ap-7678	234	8	where	where	SCONJ
ap-7678	234	9	some	some	DET
ap-7678	234	10	convenient	convenient	ADJ
ap-7678	234	11	normalization	normalization	NOUN
ap-7678	234	12	has	have	AUX
ap-7678	234	13	been	be	AUX
ap-7678	234	14	adopted	adopt	VERB
ap-7678	234	15	,	,	PUNCT
ap-7678	234	16	c(−→µ	c(−→µ	PROPN
ap-7678	234	17	)	)	PUNCT
ap-7678	234	18	=	=	PUNCT
ap-7678	234	19	1	1	NUM
ap-7678	234	20	.	.	NOUN
ap-7678	234	21	5	5	NUM
ap-7678	234	22	10	10	NUM
ap-7678	234	23	15	15	NUM
ap-7678	234	24	20	20	NUM
ap-7678	234	25	e	e	NOUN
ap-7678	234	26	1	1	NUM
ap-7678	234	27	2	2	NUM
ap-7678	234	28	3	3	NUM
ap-7678	234	29	4	4	NUM
ap-7678	234	30	5	5	NUM
ap-7678	234	31	log	log	NOUN
ap-7678	234	32	10	10	NUM
ap-7678	234	33	(	(	PUNCT
ap-7678	234	34	�	�	NOUN
ap-7678	234	35	n(e	n(e	NOUN
ap-7678	234	36	)	)	PUNCT
ap-7678	234	37	)	)	PUNCT
ap-7678	234	38	figure	figure	NOUN
ap-7678	234	39	2	2	NUM
ap-7678	234	40	.	.	X
ap-7678	234	41	log10(λn	log10(λn	PROPN
ap-7678	234	42	(	(	PUNCT
ap-7678	234	43	e	e	NOUN
ap-7678	234	44	)	)	PUNCT
ap-7678	234	45	)	)	PUNCT
ap-7678	234	46	for	for	ADP
ap-7678	234	47	sextic	sextic	ADJ
ap-7678	234	48	anharmonic	anharmonic	ADJ
ap-7678	234	49	oscillator	oscillator	NOUN
ap-7678	234	50	even	even	ADV
ap-7678	234	51	states	state	NOUN
ap-7678	234	52	in	in	ADP
ap-7678	234	53	eq	eq	ADP
ap-7678	234	54	.	.	PUNCT
ap-7678	235	1	(	(	PUNCT
ap-7678	235	2	65	65	NUM
ap-7678	235	3	)	)	PUNCT
ap-7678	235	4	;	;	PUNCT
ap-7678	236	1	n	n	PROPN
ap-7678	236	2	=	=	SYM
ap-7678	236	3	4	4	NUM
ap-7678	236	4	,	,	PUNCT
ap-7678	236	5	6	6	NUM
ap-7678	236	6	,	,	PUNCT
ap-7678	236	7	8	8	NUM
ap-7678	236	8	,	,	PUNCT
ap-7678	236	9	10	10	NUM
ap-7678	236	10	,	,	PUNCT
ap-7678	236	11	12	12	NUM
ap-7678	236	12	;	;	PUNCT
ap-7678	236	13	based	base	VERB
ap-7678	236	14	on	on	ADP
ap-7678	236	15	the	the	DET
ap-7678	236	16	weight	weight	NOUN
ap-7678	236	17	rl(ξ	rl(ξ	PUNCT
ap-7678	236	18	)	)	PUNCT
ap-7678	236	19	=	=	SYM
ap-7678	237	1	ξ−	ξ−	NOUN
ap-7678	237	2	1	1	NUM
ap-7678	237	3	2	2	NUM
ap-7678	237	4	exp(−ξ	exp(−ξ	PROPN
ap-7678	237	5	)	)	PUNCT
ap-7678	237	6	.	.	PUNCT
ap-7678	238	1	5	5	NUM
ap-7678	238	2	10	10	NUM
ap-7678	238	3	15	15	NUM
ap-7678	238	4	20	20	NUM
ap-7678	238	5	e	e	NOUN
ap-7678	238	6	2	2	NUM
ap-7678	238	7	4	4	NUM
ap-7678	238	8	6	6	NUM
ap-7678	238	9	8	8	NUM
ap-7678	238	10	10	10	NUM
ap-7678	238	11	log	log	NOUN
ap-7678	238	12	10	10	NUM
ap-7678	238	13	(	(	PUNCT
ap-7678	238	14	�	�	NOUN
ap-7678	238	15	n(e	n(e	NOUN
ap-7678	238	16	)	)	PUNCT
ap-7678	238	17	)	)	PUNCT
ap-7678	238	18	figure	figure	NOUN
ap-7678	238	19	3	3	NUM
ap-7678	238	20	.	.	X
ap-7678	238	21	log10(λn	log10(λn	PROPN
ap-7678	238	22	(	(	PUNCT
ap-7678	238	23	e	e	NOUN
ap-7678	238	24	)	)	PUNCT
ap-7678	238	25	)	)	PUNCT
ap-7678	238	26	for	for	ADP
ap-7678	238	27	sextic	sextic	ADJ
ap-7678	238	28	anharmonic	anharmonic	ADJ
ap-7678	238	29	oscillator	oscillator	NOUN
ap-7678	238	30	even	even	ADV
ap-7678	238	31	states	state	NOUN
ap-7678	238	32	in	in	ADP
ap-7678	238	33	eq	eq	ADP
ap-7678	238	34	.	.	PUNCT
ap-7678	239	1	(	(	PUNCT
ap-7678	239	2	65	65	NUM
ap-7678	239	3	)	)	PUNCT
ap-7678	239	4	;	;	PUNCT
ap-7678	240	1	n	n	PROPN
ap-7678	240	2	=	=	SYM
ap-7678	240	3	4	4	NUM
ap-7678	240	4	,	,	PUNCT
ap-7678	240	5	6	6	NUM
ap-7678	240	6	,	,	PUNCT
ap-7678	240	7	8	8	NUM
ap-7678	240	8	,	,	PUNCT
ap-7678	240	9	10	10	NUM
ap-7678	240	10	,	,	PUNCT
ap-7678	240	11	12	12	NUM
ap-7678	240	12	;	;	PUNCT
ap-7678	240	13	based	base	VERB
ap-7678	240	14	on	on	ADP
ap-7678	240	15	the	the	DET
ap-7678	240	16	weight	weight	NOUN
ap-7678	240	17	rg(ξ	rg(ξ	NOUN
ap-7678	240	18	)	)	PUNCT
ap-7678	241	1	=	=	SYM
ap-7678	241	2	ξ−	ξ−	NOUN
ap-7678	241	3	1	1	NUM
ap-7678	241	4	2	2	NUM
ap-7678	241	5	exp(−ξ2/2	exp(−ξ2/2	PRON
ap-7678	241	6	)	)	PUNCT
ap-7678	241	7	.	.	PUNCT
ap-7678	242	1	an	an	DET
ap-7678	242	2	important	important	ADJ
ap-7678	242	3	result	result	NOUN
ap-7678	242	4	is	be	AUX
ap-7678	242	5	that	that	SCONJ
ap-7678	242	6	ln	ln	ADJ
ap-7678	242	7	(	(	PUNCT
ap-7678	242	8	e	e	NOUN
ap-7678	242	9	)	)	PUNCT
ap-7678	242	10	<	<	X
ap-7678	242	11	ln+1(e	ln+1(e	PROPN
ap-7678	242	12	)	)	PUNCT
ap-7678	242	13	.	.	PUNCT
ap-7678	243	1	(	(	PUNCT
ap-7678	243	2	35	35	NUM
ap-7678	243	3	)	)	PUNCT
ap-7678	243	4	this	this	PRON
ap-7678	243	5	trivially	trivially	ADV
ap-7678	243	6	follows	follow	VERB
ap-7678	243	7	from	from	ADP
ap-7678	243	8	eqs	eqs	PROPN
ap-7678	243	9	.	.	PUNCT
ap-7678	244	1	(	(	PUNCT
ap-7678	244	2	33	33	NUM
ap-7678	244	3	-	-	SYM
ap-7678	244	4	34	34	NUM
ap-7678	244	5	)	)	PUNCT
ap-7678	244	6	.	.	PUNCT
ap-7678	245	1	an	an	DET
ap-7678	245	2	immediate	immediate	ADJ
ap-7678	245	3	consequence	consequence	NOUN
ap-7678	245	4	is	be	AUX
ap-7678	245	5	that	that	SCONJ
ap-7678	245	6	the	the	DET
ap-7678	245	7	counterpart	counterpart	NOUN
ap-7678	245	8	to	to	ADP
ap-7678	245	9	the	the	DET
ap-7678	245	10	quantization	quantization	NOUN
ap-7678	245	11	condition	condition	NOUN
ap-7678	245	12	in	in	ADP
ap-7678	245	13	eq	eq	ADP
ap-7678	245	14	.	.	PUNCT
ap-7678	246	1	(	(	PUNCT
ap-7678	246	2	22	22	NUM
ap-7678	246	3	)	)	PUNCT
ap-7678	246	4	now	now	ADV
ap-7678	246	5	becomes	become	VERB
ap-7678	246	6	simpler	simple	ADJ
ap-7678	246	7	:	:	PUNCT
ap-7678	246	8	lim	lim	PROPN
ap-7678	246	9	n→∞	n→∞	X
ap-7678	246	10	ln	ln	NOUN
ap-7678	246	11	(	(	PUNCT
ap-7678	246	12	e	e	NOUN
ap-7678	246	13	)	)	PUNCT
ap-7678	247	1	=	=	SYM
ap-7678	247	2	{	{	PUNCT
ap-7678	247	3	finite	finite	VERB
ap-7678	247	4	⇐	⇐	ADJ
ap-7678	247	5	⇒	⇒	NOUN
ap-7678	247	6	e	e	NOUN
ap-7678	247	7	=	=	SYM
ap-7678	247	8	ephys	ephys	PROPN
ap-7678	247	9	,	,	PUNCT
ap-7678	247	10	∞	∞	NUM
ap-7678	247	11	⇐	⇐	ADJ
ap-7678	247	12	⇒	⇒	NOUN
ap-7678	247	13	e	e	PROPN
ap-7678	247	14	̸=	̸=	PROPN
ap-7678	247	15	ephys	ephy	NOUN
ap-7678	247	16	.	.	PUNCT
ap-7678	248	1	(	(	PUNCT
ap-7678	248	2	36	36	NUM
ap-7678	248	3	)	)	PUNCT
ap-7678	248	4	combining	combine	VERB
ap-7678	248	5	this	this	PRON
ap-7678	248	6	with	with	ADP
ap-7678	248	7	eq	eq	NOUN
ap-7678	248	8	.	.	PUNCT
ap-7678	249	1	(	(	PUNCT
ap-7678	249	2	35	35	NUM
ap-7678	249	3	)	)	PUNCT
ap-7678	249	4	we	we	PRON
ap-7678	249	5	obtain	obtain	VERB
ap-7678	249	6	:	:	PUNCT
ap-7678	249	7	0	0	PUNCT
ap-7678	250	1	<	<	X
ap-7678	250	2	ln	ln	X
ap-7678	250	3	(	(	PUNCT
ap-7678	250	4	e	e	NOUN
ap-7678	250	5	)	)	PUNCT
ap-7678	250	6	<	<	X
ap-7678	250	7	ln+1(e	ln+1(e	PROPN
ap-7678	250	8	)	)	PUNCT
ap-7678	250	9	<	<	X
ap-7678	250	10	.	.	PUNCT
ap-7678	250	11	.	.	PUNCT
ap-7678	250	12	.	.	PUNCT
ap-7678	250	13	{	{	PUNCT
ap-7678	251	1	finite	finite	PROPN
ap-7678	251	2	,	,	PUNCT
ap-7678	251	3	⇐	⇐	ADJ
ap-7678	251	4	⇒	⇒	NOUN
ap-7678	251	5	e	e	NOUN
ap-7678	251	6	=	=	SYM
ap-7678	251	7	ephys	ephys	PROPN
ap-7678	251	8	∞	∞	PROPN
ap-7678	251	9	,	,	PUNCT
ap-7678	251	10	⇐	⇐	ADJ
ap-7678	251	11	⇒	⇒	NOUN
ap-7678	251	12	e	e	PROPN
ap-7678	251	13	̸=	̸=	PROPN
ap-7678	251	14	ephys	ephy	NOUN
ap-7678	251	15	.	.	PUNCT
ap-7678	252	1	(	(	PUNCT
ap-7678	252	2	37	37	NUM
ap-7678	252	3	)	)	PUNCT
ap-7678	252	4	therefore	therefore	ADV
ap-7678	252	5	,	,	PUNCT
ap-7678	252	6	the	the	DET
ap-7678	252	7	ln	ln	ADJ
ap-7678	252	8	(	(	PUNCT
ap-7678	252	9	e	e	NOUN
ap-7678	252	10	)	)	PUNCT
ap-7678	252	11	functions	function	NOUN
ap-7678	252	12	form	form	VERB
ap-7678	252	13	an	an	DET
ap-7678	252	14	increasing	increase	VERB
ap-7678	252	15	,	,	PUNCT
ap-7678	252	16	nested	nest	VERB
ap-7678	252	17	,	,	PUNCT
ap-7678	252	18	concaved	concave	VERB
ap-7678	252	19	upwards	upwards	ADV
ap-7678	252	20	,	,	PUNCT
ap-7678	252	21	sequence	sequence	NOUN
ap-7678	252	22	of	of	ADP
ap-7678	252	23	positive	positive	ADJ
ap-7678	252	24	functions	function	NOUN
ap-7678	252	25	.	.	PUNCT
ap-7678	253	1	this	this	PRON
ap-7678	253	2	is	be	AUX
ap-7678	253	3	demonstrated	demonstrate	VERB
ap-7678	253	4	for	for	ADP
ap-7678	253	5	the	the	DET
ap-7678	253	6	sextic	sextic	ADJ
ap-7678	253	7	anharmonic	anharmonic	ADJ
ap-7678	253	8	harmonic	harmonic	ADJ
ap-7678	253	9	oscillator	oscillator	NOUN
ap-7678	253	10	problem	problem	NOUN
ap-7678	253	11	(	(	PUNCT
ap-7678	253	12	figures	figure	NOUN
ap-7678	253	13	1	1	NUM
ap-7678	253	14	-	-	SYM
ap-7678	253	15	3	3	NUM
ap-7678	253	16	)	)	PUNCT
ap-7678	253	17	,	,	PUNCT
ap-7678	253	18	as	as	ADV
ap-7678	253	19	well	well	ADV
ap-7678	253	20	as	as	ADP
ap-7678	253	21	the	the	DET
ap-7678	253	22	non	non	ADJ
ap-7678	253	23	-	-	ADJ
ap-7678	253	24	hermitian	hermitian	ADJ
ap-7678	253	25	ix3	ix3	PROPN
ap-7678	253	26	+	+	CCONJ
ap-7678	253	27	iax	iax	PROPN
ap-7678	253	28	potential	potential	NOUN
ap-7678	253	29	(	(	PUNCT
ap-7678	253	30	figure	figure	NOUN
ap-7678	253	31	4	4	NUM
ap-7678	253	32	)	)	PUNCT
ap-7678	253	33	,	,	PUNCT
ap-7678	253	34	on	on	ADP
ap-7678	253	35	the	the	DET
ap-7678	253	36	complex	complex	ADJ
ap-7678	253	37	energy	energy	NOUN
ap-7678	253	38	domain	domain	NOUN
ap-7678	253	39	.	.	PUNCT
ap-7678	254	1	only	only	ADV
ap-7678	254	2	at	at	ADP
ap-7678	254	3	the	the	DET
ap-7678	254	4	exact	exact	ADJ
ap-7678	254	5	physical	physical	ADJ
ap-7678	254	6	energy	energy	NOUN
ap-7678	254	7	will	will	AUX
ap-7678	254	8	the	the	DET
ap-7678	254	9	limit	limit	NOUN
ap-7678	254	10	be	be	AUX
ap-7678	254	11	finite	finite	ADJ
ap-7678	254	12	.	.	PUNCT
ap-7678	255	1	everywhere	everywhere	ADV
ap-7678	255	2	else	else	ADV
ap-7678	255	3	it	it	PRON
ap-7678	255	4	will	will	AUX
ap-7678	255	5	become	become	VERB
ap-7678	255	6	infinite	infinite	ADJ
ap-7678	255	7	.	.	PUNCT
ap-7678	256	1	68	68	NUM
ap-7678	256	2	vol	vol	NOUN
ap-7678	256	3	.	.	PUNCT
ap-7678	257	1	62	62	NUM
ap-7678	257	2	no	no	INTJ
ap-7678	257	3	.	.	PUNCT
ap-7678	258	1	1/2022	1/2022	NUM
ap-7678	258	2	algebraic	algebraic	ADJ
ap-7678	258	3	eigenenergy	eigenenergy	NOUN
ap-7678	258	4	bounding	bounding	NOUN
ap-7678	258	5	method	method	NOUN
ap-7678	258	6	figure	figure	NOUN
ap-7678	258	7	4	4	NUM
ap-7678	258	8	.	.	X
ap-7678	259	1	log10(λn	log10(λn	PROPN
ap-7678	259	2	(	(	PUNCT
ap-7678	259	3	e	e	NOUN
ap-7678	259	4	)	)	PUNCT
ap-7678	259	5	)	)	PUNCT
ap-7678	260	1	for	for	ADP
ap-7678	260	2	n	n	NOUN
ap-7678	260	3	=	=	SYM
ap-7678	260	4	20	20	NUM
ap-7678	260	5	,	,	PUNCT
ap-7678	260	6	30	30	NUM
ap-7678	260	7	,	,	PUNCT
ap-7678	260	8	40	40	NUM
ap-7678	260	9	,	,	PUNCT
ap-7678	260	10	50	50	NUM
ap-7678	260	11	,	,	PUNCT
ap-7678	260	12	v	v	NOUN
ap-7678	260	13	(	(	PUNCT
ap-7678	260	14	x	x	NOUN
ap-7678	260	15	)	)	PUNCT
ap-7678	260	16	=	=	SYM
ap-7678	260	17	ix3	ix3	PROPN
ap-7678	260	18	+	+	CCONJ
ap-7678	260	19	iax	iax	PROPN
ap-7678	260	20	,	,	PUNCT
ap-7678	260	21	a	a	PRON
ap-7678	260	22	=	=	X
ap-7678	260	23	−2.70	−2.70	AUX
ap-7678	260	24	given	give	VERB
ap-7678	260	25	the	the	DET
ap-7678	260	26	behavior	behavior	NOUN
ap-7678	260	27	of	of	ADP
ap-7678	260	28	the	the	DET
ap-7678	260	29	ln	ln	ADJ
ap-7678	260	30	(	(	PUNCT
ap-7678	260	31	e	e	NOUN
ap-7678	260	32	)	)	PUNCT
ap-7678	260	33	,	,	PUNCT
ap-7678	260	34	it	it	PRON
ap-7678	260	35	is	be	AUX
ap-7678	260	36	clear	clear	ADJ
ap-7678	260	37	that	that	SCONJ
ap-7678	260	38	the	the	DET
ap-7678	260	39	local	local	ADJ
ap-7678	260	40	minima	minima	NOUN
ap-7678	260	41	in	in	ADP
ap-7678	260	42	the	the	DET
ap-7678	260	43	energy	energy	NOUN
ap-7678	260	44	variable	variable	NOUN
ap-7678	260	45	,	,	PUNCT
ap-7678	260	46	at	at	ADP
ap-7678	260	47	a	a	DET
ap-7678	260	48	given	give	VERB
ap-7678	260	49	order	order	NOUN
ap-7678	260	50	“	"	PUNCT
ap-7678	260	51	n	n	CCONJ
ap-7678	260	52	”	"	PUNCT
ap-7678	260	53	,	,	PUNCT
ap-7678	260	54	should	should	AUX
ap-7678	260	55	approximate	approximate	VERB
ap-7678	260	56	the	the	DET
ap-7678	260	57	physical	physical	ADJ
ap-7678	260	58	energies	energy	NOUN
ap-7678	260	59	.	.	PUNCT
ap-7678	261	1	thus	thus	ADV
ap-7678	261	2	,	,	PUNCT
ap-7678	261	3	for	for	ADP
ap-7678	261	4	each	each	DET
ap-7678	261	5	physical	physical	ADJ
ap-7678	261	6	discrete	discrete	ADJ
ap-7678	261	7	state	state	NOUN
ap-7678	261	8	,	,	PUNCT
ap-7678	261	9	its	its	PRON
ap-7678	261	10	corresponding	correspond	VERB
ap-7678	261	11	approximants	approximant	NOUN
ap-7678	261	12	,	,	PUNCT
ap-7678	261	13	ephys;n	ephys;n	VERB
ap-7678	261	14	,	,	PUNCT
ap-7678	261	15	will	will	AUX
ap-7678	261	16	satisfy	satisfy	VERB
ap-7678	261	17	∂eln	∂eln	PUNCT
ap-7678	261	18	(	(	PUNCT
ap-7678	261	19	e(min	e(min	NOUN
ap-7678	261	20	)	)	PUNCT
ap-7678	261	21	phys;n	phys;n	PROPN
ap-7678	261	22	)	)	PUNCT
ap-7678	262	1	=	=	SYM
ap-7678	262	2	0	0	NUM
ap-7678	262	3	,	,	PUNCT
ap-7678	262	4	(	(	PUNCT
ap-7678	262	5	38	38	NUM
ap-7678	262	6	)	)	PUNCT
ap-7678	262	7	resulting	result	VERB
ap-7678	262	8	in	in	ADP
ap-7678	262	9	lim	lim	PROPN
ap-7678	262	10	n→∞	n→∞	NUM
ap-7678	262	11	e	e	X
ap-7678	262	12	(	(	PUNCT
ap-7678	262	13	min	min	NOUN
ap-7678	262	14	)	)	PUNCT
ap-7678	262	15	phys;n	phys;n	NOUN
ap-7678	262	16	=	=	SYM
ap-7678	262	17	ephys	ephys	PROPN
ap-7678	262	18	.	.	PUNCT
ap-7678	263	1	(	(	PUNCT
ap-7678	263	2	39	39	NUM
ap-7678	263	3	)	)	PUNCT
ap-7678	263	4	more	more	ADV
ap-7678	263	5	importantly	importantly	ADV
ap-7678	263	6	,	,	PUNCT
ap-7678	263	7	these	these	DET
ap-7678	263	8	local	local	ADJ
ap-7678	263	9	minima	minima	NOUN
ap-7678	263	10	have	have	VERB
ap-7678	263	11	a	a	DET
ap-7678	263	12	very	very	ADV
ap-7678	263	13	important	important	ADJ
ap-7678	263	14	property	property	NOUN
ap-7678	263	15	.	.	PUNCT
ap-7678	264	1	the	the	DET
ap-7678	264	2	expressions	expression	NOUN
ap-7678	264	3	ln	ln	X
ap-7678	264	4	(	(	PUNCT
ap-7678	264	5	e(min	e(min	NOUN
ap-7678	264	6	)	)	PUNCT
ap-7678	264	7	phys;n	phys;n	PROPN
ap-7678	264	8	)	)	PUNCT
ap-7678	264	9	form	form	VERB
ap-7678	264	10	an	an	DET
ap-7678	264	11	increasing	increase	VERB
ap-7678	264	12	,	,	PUNCT
ap-7678	264	13	positive	positive	ADJ
ap-7678	264	14	sequence	sequence	NOUN
ap-7678	264	15	,	,	PUNCT
ap-7678	264	16	bounded	bound	VERB
ap-7678	264	17	from	from	ADP
ap-7678	264	18	above	above	ADV
ap-7678	264	19	by	by	ADP
ap-7678	264	20	the	the	DET
ap-7678	264	21	physical	physical	ADJ
ap-7678	264	22	counterpart	counterpart	NOUN
ap-7678	264	23	.	.	PUNCT
ap-7678	265	1	this	this	PRON
ap-7678	265	2	follows	follow	VERB
ap-7678	265	3	from	from	ADP
ap-7678	265	4	ln	ln	ADJ
ap-7678	265	5	(	(	PUNCT
ap-7678	265	6	e(min	e(min	PROPN
ap-7678	265	7	)	)	PUNCT
ap-7678	265	8	phys;n	phys;n	PROPN
ap-7678	265	9	)	)	PUNCT
ap-7678	265	10	<	<	X
ap-7678	265	11	ln+1(e(min	ln+1(e(min	X
ap-7678	265	12	)	)	PUNCT
ap-7678	265	13	phys;n+1	phys;n+1	PROPN
ap-7678	265	14	)	)	PUNCT
ap-7678	265	15	,	,	PUNCT
ap-7678	265	16	(	(	PUNCT
ap-7678	265	17	40	40	NUM
ap-7678	265	18	)	)	PUNCT
ap-7678	265	19	which	which	PRON
ap-7678	265	20	follows	follow	VERB
ap-7678	265	21	from	from	ADP
ap-7678	265	22	:	:	PUNCT
ap-7678	265	23	ln	ln	ADJ
ap-7678	265	24	(	(	PUNCT
ap-7678	265	25	e(min	e(min	PROPN
ap-7678	265	26	)	)	PUNCT
ap-7678	265	27	phys;n	phys;n	PROPN
ap-7678	265	28	)	)	PUNCT
ap-7678	265	29	<	<	X
ap-7678	265	30	ln	ln	X
ap-7678	265	31	(	(	PUNCT
ap-7678	265	32	e(min	e(min	PROPN
ap-7678	265	33	)	)	PUNCT
ap-7678	265	34	phys;n+1	phys;n+1	NOUN
ap-7678	265	35	)	)	PUNCT
ap-7678	265	36	<	<	X
ap-7678	265	37	ln+1(e(min	ln+1(e(min	X
ap-7678	265	38	)	)	PUNCT
ap-7678	265	39	phys;n+1	phys;n+1	PROPN
ap-7678	265	40	)	)	PUNCT
ap-7678	265	41	.	.	PUNCT
ap-7678	266	1	(	(	PUNCT
ap-7678	266	2	41	41	NUM
ap-7678	266	3	)	)	PUNCT
ap-7678	266	4	it	it	PRON
ap-7678	266	5	now	now	ADV
ap-7678	266	6	follows	follow	VERB
ap-7678	266	7	that	that	SCONJ
ap-7678	266	8	the	the	DET
ap-7678	266	9	counterpart	counterpart	NOUN
ap-7678	266	10	to	to	ADP
ap-7678	266	11	the	the	DET
ap-7678	266	12	oppq	oppq	ADJ
ap-7678	266	13	quantization	quantization	NOUN
ap-7678	266	14	condition	condition	NOUN
ap-7678	266	15	in	in	ADP
ap-7678	266	16	eq	eq	ADP
ap-7678	266	17	.	.	PUNCT
ap-7678	267	1	(	(	PUNCT
ap-7678	267	2	22	22	NUM
ap-7678	267	3	)	)	PUNCT
ap-7678	267	4	becomes	become	VERB
ap-7678	267	5	0	0	NUM
ap-7678	267	6	<	<	X
ap-7678	267	7	ln	ln	X
ap-7678	267	8	(	(	PUNCT
ap-7678	267	9	e(min	e(min	PROPN
ap-7678	267	10	)	)	PUNCT
ap-7678	267	11	phys;n	phys;n	PROPN
ap-7678	267	12	)	)	PUNCT
ap-7678	267	13	<	<	X
ap-7678	267	14	ln+1(e(min	ln+1(e(min	X
ap-7678	267	15	)	)	PUNCT
ap-7678	267	16	phys;n+1	phys;n+1	PROPN
ap-7678	267	17	)	)	PUNCT
ap-7678	267	18	<	<	X
ap-7678	267	19	.	.	PUNCT
ap-7678	267	20	.	.	PUNCT
ap-7678	267	21	.	.	PUNCT
ap-7678	268	1	<	<	X
ap-7678	268	2	l∞(ephys	l∞(ephys	PROPN
ap-7678	268	3	)	)	PUNCT
ap-7678	268	4	.	.	PUNCT
ap-7678	269	1	(	(	PUNCT
ap-7678	269	2	42	42	NUM
ap-7678	269	3	)	)	PUNCT
ap-7678	269	4	thus	thus	ADV
ap-7678	269	5	,	,	PUNCT
ap-7678	269	6	the	the	DET
ap-7678	269	7	{	{	PUNCT
ap-7678	269	8	ln	ln	ADJ
ap-7678	269	9	(	(	PUNCT
ap-7678	269	10	e(min	e(min	NOUN
ap-7678	269	11	)	)	PUNCT
ap-7678	269	12	phys;n	phys;n	NOUN
ap-7678	269	13	)	)	PUNCT
ap-7678	269	14	}	}	PUNCT
ap-7678	269	15	form	form	VERB
ap-7678	269	16	a	a	DET
ap-7678	269	17	monotonically	monotonically	ADV
ap-7678	269	18	increasing	increase	VERB
ap-7678	269	19	positive	positive	ADJ
ap-7678	269	20	sequence	sequence	NOUN
ap-7678	269	21	bounded	bound	VERB
ap-7678	269	22	from	from	ADP
ap-7678	269	23	above	above	ADV
ap-7678	269	24	by	by	ADP
ap-7678	269	25	the	the	DET
ap-7678	269	26	physical	physical	ADJ
ap-7678	269	27	value	value	NOUN
ap-7678	269	28	.	.	PUNCT
ap-7678	270	1	the	the	DET
ap-7678	270	2	minima	minima	NOUN
ap-7678	270	3	do	do	AUX
ap-7678	270	4	not	not	PART
ap-7678	270	5	necessarily	necessarily	ADV
ap-7678	270	6	converge	converge	VERB
ap-7678	270	7	monotonically	monotonically	ADV
ap-7678	270	8	.	.	PUNCT
ap-7678	271	1	8	8	X
ap-7678	271	2	.	.	PUNCT
ap-7678	272	1	the	the	DET
ap-7678	272	2	oppq	oppq	NOUN
ap-7678	272	3	-	-	PUNCT
ap-7678	272	4	bounding	bound	VERB
ap-7678	272	5	method	method	NOUN
ap-7678	272	6	upon	upon	SCONJ
ap-7678	272	7	reviewing	review	VERB
ap-7678	272	8	eq	eq	ADP
ap-7678	272	9	.	.	PUNCT
ap-7678	273	1	(	(	PUNCT
ap-7678	273	2	42	42	NUM
ap-7678	273	3	)	)	PUNCT
ap-7678	273	4	a	a	DET
ap-7678	273	5	bounding	bounding	NOUN
ap-7678	273	6	strategy	strategy	NOUN
ap-7678	273	7	emerges	emerge	VERB
ap-7678	273	8	.	.	PUNCT
ap-7678	274	1	assume	assume	VERB
ap-7678	274	2	that	that	SCONJ
ap-7678	274	3	the	the	DET
ap-7678	274	4	sequence	sequence	NOUN
ap-7678	274	5	elements	element	NOUN
ap-7678	274	6	can	can	AUX
ap-7678	274	7	be	be	AUX
ap-7678	274	8	generated	generate	VERB
ap-7678	274	9	to	to	ADP
ap-7678	274	10	sufficiently	sufficiently	ADV
ap-7678	274	11	high	high	ADJ
ap-7678	274	12	expansion	expansion	NOUN
ap-7678	274	13	orders	order	NOUN
ap-7678	274	14	,	,	PUNCT
ap-7678	274	15	and	and	CCONJ
ap-7678	274	16	that	that	SCONJ
ap-7678	274	17	a	a	DET
ap-7678	274	18	rough	rough	ADJ
ap-7678	274	19	upper	upper	ADJ
ap-7678	274	20	bound	bind	VERB
ap-7678	274	21	,	,	PUNCT
ap-7678	274	22	bu	bu	PROPN
ap-7678	274	23	can	can	AUX
ap-7678	274	24	be	be	AUX
ap-7678	274	25	discerned	discern	VERB
ap-7678	274	26	:	:	PUNCT
ap-7678	274	27	ln	ln	ADJ
ap-7678	274	28	(	(	PUNCT
ap-7678	274	29	e(min	e(min	PROPN
ap-7678	274	30	)	)	PUNCT
ap-7678	274	31	phys;n	phys;n	PROPN
ap-7678	274	32	)	)	PUNCT
ap-7678	274	33	<	<	X
ap-7678	274	34	ln+1(e(min	ln+1(e(min	X
ap-7678	274	35	)	)	PUNCT
ap-7678	274	36	phys;n+1	phys;n+1	PROPN
ap-7678	274	37	)	)	PUNCT
ap-7678	274	38	<	<	X
ap-7678	274	39	.	.	PUNCT
ap-7678	274	40	.	.	PUNCT
ap-7678	274	41	.	.	PUNCT
ap-7678	275	1	<	<	X
ap-7678	275	2	l∞(ephys	l∞(ephys	PROPN
ap-7678	275	3	)	)	PUNCT
ap-7678	275	4	<	<	X
ap-7678	275	5	bu	bu	INTJ
ap-7678	275	6	.	.	PUNCT
ap-7678	276	1	(	(	PUNCT
ap-7678	276	2	43	43	NUM
ap-7678	276	3	)	)	PUNCT
ap-7678	276	4	given	give	VERB
ap-7678	276	5	the	the	DET
ap-7678	276	6	behavior	behavior	NOUN
ap-7678	276	7	of	of	ADP
ap-7678	276	8	the	the	DET
ap-7678	276	9	ln	ln	ADJ
ap-7678	276	10	(	(	PUNCT
ap-7678	276	11	e	e	NOUN
ap-7678	276	12	)	)	PUNCT
ap-7678	276	13	functions	function	NOUN
ap-7678	276	14	,	,	PUNCT
ap-7678	276	15	as	as	SCONJ
ap-7678	276	16	given	give	VERB
ap-7678	276	17	in	in	ADP
ap-7678	276	18	eqs	eqs	PROPN
ap-7678	276	19	.	.	PUNCT
ap-7678	277	1	(	(	PUNCT
ap-7678	277	2	35	35	NUM
ap-7678	277	3	-	-	SYM
ap-7678	277	4	37	37	NUM
ap-7678	277	5	)	)	PUNCT
ap-7678	277	6	,	,	PUNCT
ap-7678	277	7	one	one	PRON
ap-7678	277	8	can	can	AUX
ap-7678	277	9	readily	readily	ADV
ap-7678	277	10	determine	determine	VERB
ap-7678	277	11	energy	energy	NOUN
ap-7678	277	12	parameter	parameter	NOUN
ap-7678	277	13	values	value	NOUN
ap-7678	277	14	satisfying	satisfy	VERB
ap-7678	277	15	ln	ln	ADV
ap-7678	277	16	(	(	PUNCT
ap-7678	277	17	e(l	e(l	PROPN
ap-7678	277	18	)	)	PUNCT
ap-7678	277	19	phys;n	phys;n	PROPN
ap-7678	277	20	)	)	PUNCT
ap-7678	278	1	=	=	SYM
ap-7678	278	2	ln	ln	ADJ
ap-7678	278	3	(	(	PUNCT
ap-7678	278	4	e(u	e(u	PROPN
ap-7678	278	5	)	)	PUNCT
ap-7678	278	6	phys;n	phys;n	PROPN
ap-7678	278	7	)	)	PUNCT
ap-7678	279	1	=	=	SYM
ap-7678	279	2	bu	bu	INTJ
ap-7678	279	3	,	,	PUNCT
ap-7678	279	4	(	(	PUNCT
ap-7678	279	5	44	44	NUM
ap-7678	279	6	)	)	PUNCT
ap-7678	279	7	such	such	ADJ
ap-7678	279	8	that	that	SCONJ
ap-7678	279	9	e	e	NOUN
ap-7678	279	10	(	(	PUNCT
ap-7678	279	11	l	l	NOUN
ap-7678	279	12	)	)	PUNCT
ap-7678	279	13	phys;n	phys;n	PROPN
ap-7678	279	14	<	<	X
ap-7678	279	15	ephys	ephys	X
ap-7678	279	16	<	<	X
ap-7678	279	17	e	e	X
ap-7678	279	18	(	(	PUNCT
ap-7678	279	19	u	u	NOUN
ap-7678	279	20	)	)	PUNCT
ap-7678	279	21	phys;n	phys;n	PROPN
ap-7678	279	22	,	,	PUNCT
ap-7678	279	23	(	(	PUNCT
ap-7678	279	24	45	45	NUM
ap-7678	279	25	)	)	PUNCT
ap-7678	279	26	and	and	CCONJ
ap-7678	279	27	lim	lim	PROPN
ap-7678	279	28	n→∞	n→∞	X
ap-7678	279	29	(	(	PUNCT
ap-7678	279	30	e	e	X
ap-7678	279	31	(	(	PUNCT
ap-7678	279	32	u	u	NOUN
ap-7678	279	33	)	)	PUNCT
ap-7678	279	34	phys;n	phys;n	VERB
ap-7678	279	35	−	−	PROPN
ap-7678	279	36	e	e	X
ap-7678	279	37	(	(	PUNCT
ap-7678	279	38	l	l	NOUN
ap-7678	279	39	)	)	PUNCT
ap-7678	279	40	phys;n	phys;n	PROPN
ap-7678	279	41	)	)	PUNCT
ap-7678	280	1	=	=	PUNCT
ap-7678	280	2	0	0	PUNCT
ap-7678	281	1	+	+	NOUN
ap-7678	281	2	.	.	PUNCT
ap-7678	282	1	(	(	PUNCT
ap-7678	282	2	46	46	NUM
ap-7678	282	3	)	)	PUNCT
ap-7678	282	4	as	as	ADP
ap-7678	282	5	a	a	DET
ap-7678	282	6	point	point	NOUN
ap-7678	282	7	of	of	ADP
ap-7678	282	8	comparison	comparison	NOUN
ap-7678	282	9	,	,	PUNCT
ap-7678	282	10	the	the	DET
ap-7678	282	11	rayleigh	rayleigh	PROPN
ap-7678	282	12	-	-	PUNCT
ap-7678	282	13	ritz	ritz	PROPN
ap-7678	282	14	(	(	PUNCT
ap-7678	282	15	rr	rr	NOUN
ap-7678	282	16	)	)	PUNCT
ap-7678	282	17	method	method	NOUN
ap-7678	282	18	solely	solely	ADV
ap-7678	282	19	produces	produce	VERB
ap-7678	282	20	upper	upper	ADJ
ap-7678	282	21	bounds	bound	NOUN
ap-7678	282	22	.	.	PUNCT
ap-7678	283	1	regardless	regardless	ADV
ap-7678	283	2	of	of	ADP
ap-7678	283	3	how	how	SCONJ
ap-7678	283	4	rapidly	rapidly	ADV
ap-7678	283	5	these	these	DET
ap-7678	283	6	bounds	bound	NOUN
ap-7678	283	7	converge	converge	VERB
ap-7678	283	8	from	from	ADP
ap-7678	283	9	above	above	ADV
ap-7678	283	10	(	(	PUNCT
ap-7678	283	11	to	to	ADP
ap-7678	283	12	the	the	DET
ap-7678	283	13	physical	physical	ADJ
ap-7678	283	14	value	value	NOUN
ap-7678	283	15	)	)	PUNCT
ap-7678	283	16	,	,	PUNCT
ap-7678	283	17	there	there	PRON
ap-7678	283	18	is	be	VERB
ap-7678	283	19	no	no	DET
ap-7678	283	20	theoretical	theoretical	ADJ
ap-7678	283	21	criteria	criterion	NOUN
ap-7678	283	22	by	by	ADP
ap-7678	283	23	which	which	PRON
ap-7678	283	24	the	the	DET
ap-7678	283	25	rr	rr	PROPN
ap-7678	283	26	results	result	NOUN
ap-7678	283	27	can	can	AUX
ap-7678	283	28	suggest	suggest	VERB
ap-7678	283	29	a	a	DET
ap-7678	283	30	lower	lower	ADV
ap-7678	283	31	bound	bind	VERB
ap-7678	283	32	to	to	ADP
ap-7678	283	33	the	the	DET
ap-7678	283	34	physical	physical	ADJ
ap-7678	283	35	energy	energy	NOUN
ap-7678	283	36	.	.	PUNCT
ap-7678	284	1	the	the	DET
ap-7678	284	2	oppq	oppq	PROPN
ap-7678	284	3	-	-	PUNCT
ap-7678	284	4	bm	bm	PROPN
ap-7678	284	5	method	method	NOUN
ap-7678	284	6	does	do	VERB
ap-7678	284	7	.	.	PUNCT
ap-7678	285	1	9	9	X
ap-7678	285	2	.	.	X
ap-7678	286	1	the	the	DET
ap-7678	286	2	sextic	sextic	PROPN
ap-7678	286	3	anharmonic	anharmonic	ADJ
ap-7678	286	4	double	double	ADJ
ap-7678	286	5	well	well	NOUN
ap-7678	286	6	potential	potential	ADJ
ap-7678	286	7	the	the	DET
ap-7678	286	8	sextic	sextic	ADJ
ap-7678	286	9	anharmonic	anharmonic	ADJ
ap-7678	286	10	oscillator	oscillator	NOUN
ap-7678	286	11	(	(	PUNCT
ap-7678	286	12	double	double	ADV
ap-7678	286	13	well	well	ADV
ap-7678	286	14	)	)	PUNCT
ap-7678	286	15	potential	potential	ADJ
ap-7678	286	16	problem	problem	NOUN
ap-7678	286	17	is	be	AUX
ap-7678	286	18	defined	define	VERB
ap-7678	286	19	by	by	ADP
ap-7678	286	20	−∂2	−∂2	PROPN
ap-7678	286	21	xψ(x	xψ(x	NUM
ap-7678	286	22	)	)	PUNCT
ap-7678	287	1	+	+	CCONJ
ap-7678	287	2	(	(	PUNCT
ap-7678	287	3	x6	x6	PROPN
ap-7678	287	4	+	+	CCONJ
ap-7678	287	5	mx2)ψ(x	mx2)ψ(x	NOUN
ap-7678	287	6	)	)	PUNCT
ap-7678	287	7	=	=	SYM
ap-7678	287	8	eψ(x	eψ(x	NOUN
ap-7678	287	9	)	)	PUNCT
ap-7678	287	10	,	,	PUNCT
ap-7678	287	11	(	(	PUNCT
ap-7678	287	12	47	47	NUM
ap-7678	287	13	)	)	PUNCT
ap-7678	287	14	where	where	SCONJ
ap-7678	287	15	we	we	PRON
ap-7678	287	16	will	will	AUX
ap-7678	287	17	take	take	VERB
ap-7678	287	18	g	g	PRON
ap-7678	287	19	≡	≡	PROPN
ap-7678	287	20	1	1	NUM
ap-7678	287	21	and	and	CCONJ
ap-7678	287	22	m	m	PROPN
ap-7678	287	23	=	=	NOUN
ap-7678	288	1	−4	−4	ADJ
ap-7678	288	2	.	.	PUNCT
ap-7678	289	1	there	there	PRON
ap-7678	289	2	are	be	VERB
ap-7678	289	3	three	three	NUM
ap-7678	289	4	different	different	ADJ
ap-7678	289	5	configuration	configuration	NOUN
ap-7678	289	6	space	space	NOUN
ap-7678	289	7	representations	representation	NOUN
ap-7678	289	8	for	for	ADP
ap-7678	289	9	the	the	DET
ap-7678	289	10	sextic	sextic	ADJ
ap-7678	289	11	anharmonic	anharmonic	ADJ
ap-7678	289	12	oscillator	oscillator	NOUN
ap-7678	289	13	problem	problem	NOUN
ap-7678	289	14	,	,	PUNCT
ap-7678	289	15	each	each	PRON
ap-7678	289	16	with	with	ADP
ap-7678	289	17	different	different	ADJ
ap-7678	289	18	mer	mer	NOUN
ap-7678	289	19	relations	relation	NOUN
ap-7678	289	20	of	of	ADP
ap-7678	289	21	varying	vary	VERB
ap-7678	289	22	order	order	NOUN
ap-7678	289	23	.	.	PUNCT
ap-7678	290	1	the	the	DET
ap-7678	290	2	most	most	ADV
ap-7678	290	3	immediate	immediate	ADJ
ap-7678	290	4	is	be	AUX
ap-7678	290	5	simply	simply	ADV
ap-7678	290	6	working	work	VERB
ap-7678	290	7	with	with	ADP
ap-7678	290	8	ψ	ψ	PRON
ap-7678	290	9	as	as	SCONJ
ap-7678	290	10	given	give	VERB
ap-7678	290	11	in	in	ADP
ap-7678	290	12	eq	eq	ADP
ap-7678	290	13	.	.	PUNCT
ap-7678	291	1	(	(	PUNCT
ap-7678	291	2	47	47	NUM
ap-7678	291	3	)	)	PUNCT
ap-7678	291	4	.	.	PUNCT
ap-7678	292	1	this	this	PRON
ap-7678	292	2	leads	lead	VERB
ap-7678	292	3	to	to	ADP
ap-7678	292	4	a	a	DET
ap-7678	292	5	sixth	sixth	ADJ
ap-7678	292	6	order	order	NOUN
ap-7678	292	7	(	(	PUNCT
ap-7678	292	8	i.e.	i.e.	X
ap-7678	292	9	1	1	NUM
ap-7678	292	10	+	+	CCONJ
ap-7678	292	11	ms	ms	NOUN
ap-7678	292	12	=	=	PROPN
ap-7678	292	13	6	6	NUM
ap-7678	292	14	)	)	PUNCT
ap-7678	292	15	finite	finite	ADJ
ap-7678	292	16	difference	difference	NOUN
ap-7678	292	17	mer	mer	PROPN
ap-7678	292	18	relation	relation	PROPN
ap-7678	292	19	,	,	PUNCT
ap-7678	292	20	as	as	SCONJ
ap-7678	292	21	given	give	VERB
ap-7678	292	22	in	in	ADP
ap-7678	292	23	eq	eq	ADP
ap-7678	292	24	.	.	PUNCT
ap-7678	293	1	(	(	PUNCT
ap-7678	293	2	48	48	NUM
ap-7678	293	3	)	)	PUNCT
ap-7678	293	4	.	.	PUNCT
ap-7678	294	1	we	we	PRON
ap-7678	294	2	examine	examine	VERB
ap-7678	294	3	both	both	PRON
ap-7678	294	4	oppq	oppq	NOUN
ap-7678	294	5	-	-	PUNCT
ap-7678	294	6	am	am	NOUN
ap-7678	294	7	and	and	CCONJ
ap-7678	294	8	oppq	oppq	NOUN
ap-7678	294	9	-	-	PUNCT
ap-7678	294	10	bm	bm	PROPN
ap-7678	294	11	as	as	SCONJ
ap-7678	294	12	applied	apply	VERB
ap-7678	294	13	to	to	ADP
ap-7678	294	14	this	this	DET
ap-7678	294	15	representation	representation	NOUN
ap-7678	294	16	.	.	PUNCT
ap-7678	295	1	the	the	DET
ap-7678	295	2	next	next	ADJ
ap-7678	295	3	configuration	configuration	NOUN
ap-7678	295	4	space	space	NOUN
ap-7678	295	5	representation	representation	NOUN
ap-7678	295	6	is	be	AUX
ap-7678	295	7	that	that	PRON
ap-7678	295	8	of	of	ADP
ap-7678	295	9	the	the	DET
ap-7678	295	10	ψ2(x	ψ2(x	NOUN
ap-7678	295	11	)	)	PUNCT
ap-7678	295	12	presentation	presentation	NOUN
ap-7678	295	13	.	.	PUNCT
ap-7678	296	1	it	it	PRON
ap-7678	296	2	leads	lead	VERB
ap-7678	296	3	to	to	ADP
ap-7678	296	4	a	a	DET
ap-7678	296	5	mer	mer	NOUN
ap-7678	296	6	relation	relation	NOUN
ap-7678	296	7	of	of	ADP
ap-7678	296	8	order	order	NOUN
ap-7678	296	9	3	3	NUM
ap-7678	296	10	(	(	PUNCT
ap-7678	296	11	i.e.	i.e.	X
ap-7678	296	12	1	1	NUM
ap-7678	296	13	+	+	CCONJ
ap-7678	296	14	ms	ms	NOUN
ap-7678	296	15	=	=	PROPN
ap-7678	296	16	3	3	NUM
ap-7678	296	17	)	)	PUNCT
ap-7678	296	18	.	.	PUNCT
ap-7678	297	1	we	we	PRON
ap-7678	297	2	do	do	AUX
ap-7678	297	3	not	not	PART
ap-7678	297	4	apply	apply	VERB
ap-7678	297	5	either	either	CCONJ
ap-7678	297	6	oppq	oppq	ADJ
ap-7678	297	7	formulation	formulation	NOUN
ap-7678	297	8	to	to	ADP
ap-7678	297	9	this	this	DET
ap-7678	297	10	case	case	NOUN
ap-7678	297	11	.	.	PUNCT
ap-7678	298	1	however	however	ADV
ap-7678	298	2	,	,	PUNCT
ap-7678	298	3	one	one	PRON
ap-7678	298	4	can	can	AUX
ap-7678	298	5	use	use	VERB
ap-7678	298	6	emm	emm	PROPN
ap-7678	298	7	to	to	PART
ap-7678	298	8	bound	bind	VERB
ap-7678	298	9	all	all	DET
ap-7678	298	10	the	the	DET
ap-7678	298	11	low	low	ADJ
ap-7678	298	12	lying	lie	VERB
ap-7678	298	13	discrete	discrete	ADJ
ap-7678	298	14	states	state	NOUN
ap-7678	298	15	;	;	PUNCT
ap-7678	298	16	thereby	thereby	ADV
ap-7678	298	17	providing	provide	VERB
ap-7678	298	18	a	a	DET
ap-7678	298	19	test	test	NOUN
ap-7678	298	20	for	for	ADP
ap-7678	298	21	the	the	DET
ap-7678	298	22	effectiveness	effectiveness	NOUN
ap-7678	298	23	of	of	ADP
ap-7678	298	24	oppq	oppq	NOUN
ap-7678	298	25	.	.	PUNCT
ap-7678	299	1	the	the	DET
ap-7678	299	2	third	third	ADJ
ap-7678	299	3	,	,	PUNCT
ap-7678	299	4	and	and	CCONJ
ap-7678	299	5	last	last	ADJ
ap-7678	299	6	,	,	PUNCT
ap-7678	299	7	sextic	sextic	ADJ
ap-7678	299	8	configuration	configuration	NOUN
ap-7678	299	9	space	space	NOUN
ap-7678	299	10	representation	representation	NOUN
ap-7678	299	11	is	be	AUX
ap-7678	299	12	provided	provide	VERB
ap-7678	299	13	by	by	ADP
ap-7678	299	14	the	the	DET
ap-7678	299	15	contact	contact	NOUN
ap-7678	299	16	transformation	transformation	NOUN
ap-7678	299	17	,	,	PUNCT
ap-7678	299	18	φ(x	φ(x	PROPN
ap-7678	299	19	)	)	PUNCT
ap-7678	299	20	=	=	X
ap-7678	299	21	exp(−	exp(−	PROPN
ap-7678	299	22	x4	x4	PROPN
ap-7678	299	23	4	4	NUM
ap-7678	299	24	)	)	PUNCT
ap-7678	299	25	ψ(x	ψ(x	NOUN
ap-7678	299	26	)	)	PUNCT
ap-7678	299	27	.	.	PUNCT
ap-7678	300	1	this	this	PRON
ap-7678	300	2	leads	lead	VERB
ap-7678	300	3	to	to	ADP
ap-7678	300	4	the	the	DET
ap-7678	300	5	most	most	ADV
ap-7678	300	6	efficient	efficient	ADJ
ap-7678	300	7	mer	mer	NOUN
ap-7678	300	8	representation	representation	NOUN
ap-7678	300	9	,	,	PUNCT
ap-7678	300	10	corresponding	correspond	VERB
ap-7678	300	11	to	to	ADP
ap-7678	300	12	a	a	DET
ap-7678	300	13	first	first	ADJ
ap-7678	300	14	order	order	NOUN
ap-7678	300	15	(	(	PUNCT
ap-7678	300	16	homogeneous	homogeneous	ADJ
ap-7678	300	17	)	)	PUNCT
ap-7678	300	18	mer	mer	PROPN
ap-7678	300	19	relation	relation	NOUN
ap-7678	300	20	(	(	PUNCT
ap-7678	300	21	i.e	i.e	PROPN
ap-7678	300	22	,	,	PUNCT
ap-7678	300	23	1	1	NUM
ap-7678	300	24	+	+	CCONJ
ap-7678	300	25	ms	ms	NOUN
ap-7678	300	26	=	=	PROPN
ap-7678	300	27	1	1	NUM
ap-7678	300	28	)	)	PUNCT
ap-7678	300	29	.	.	PUNCT
ap-7678	301	1	it	it	PRON
ap-7678	301	2	results	result	VERB
ap-7678	301	3	in	in	ADP
ap-7678	301	4	the	the	DET
ap-7678	301	5	fastest	fast	ADJ
ap-7678	301	6	oppq	oppq	ADJ
ap-7678	301	7	and	and	CCONJ
ap-7678	301	8	emm	emm	PROPN
ap-7678	301	9	convergence	convergence	NOUN
ap-7678	301	10	.	.	PUNCT
ap-7678	302	1	9.1	9.1	NUM
ap-7678	302	2	.	.	PUNCT
ap-7678	303	1	emm	emm	PROPN
ap-7678	303	2	-	-	PUNCT
ap-7678	303	3	ψ	ψ	NOUN
ap-7678	303	4	the	the	DET
ap-7678	303	5	first	first	ADJ
ap-7678	303	6	mer	mer	PROPN
ap-7678	303	7	representation	representation	NOUN
ap-7678	303	8	to	to	PART
ap-7678	303	9	be	be	AUX
ap-7678	303	10	considered	consider	VERB
ap-7678	303	11	,	,	PUNCT
ap-7678	303	12	for	for	ADP
ap-7678	303	13	the	the	DET
ap-7678	303	14	sextic	sextic	ADJ
ap-7678	303	15	anharmonic	anharmonic	ADJ
ap-7678	303	16	oscillator	oscillator	NOUN
ap-7678	303	17	,	,	PUNCT
ap-7678	303	18	results	result	VERB
ap-7678	303	19	from	from	ADP
ap-7678	303	20	a	a	DET
ap-7678	303	21	direct	direct	ADJ
ap-7678	303	22	mer	mer	NOUN
ap-7678	303	23	analysis	analysis	NOUN
ap-7678	303	24	of	of	ADP
ap-7678	303	25	the	the	DET
ap-7678	303	26	schrödinger	schrödinger	ADJ
ap-7678	303	27	equation	equation	NOUN
ap-7678	303	28	representation	representation	NOUN
ap-7678	303	29	in	in	ADP
ap-7678	303	30	eq	eq	ADP
ap-7678	303	31	.	.	PUNCT
ap-7678	304	1	(	(	PUNCT
ap-7678	304	2	47	47	NUM
ap-7678	304	3	)	)	PUNCT
ap-7678	304	4	.	.	PUNCT
ap-7678	305	1	the	the	DET
ap-7678	305	2	power	power	NOUN
ap-7678	305	3	moments	moment	NOUN
ap-7678	305	4	along	along	ADP
ap-7678	305	5	the	the	DET
ap-7678	305	6	entire	entire	ADJ
ap-7678	305	7	real	real	ADJ
ap-7678	305	8	axis	axis	NOUN
ap-7678	305	9	are	be	AUX
ap-7678	305	10	referred	refer	VERB
ap-7678	305	11	to	to	ADP
ap-7678	305	12	as	as	SCONJ
ap-7678	305	13	the	the	DET
ap-7678	305	14	hamburger	hamburger	NOUN
ap-7678	305	15	moments	moment	NOUN
ap-7678	305	16	µ(p	µ(p	PROPN
ap-7678	305	17	)	)	PUNCT
ap-7678	305	18	=	=	SYM
ap-7678	305	19	∫	∫	PROPN
ap-7678	305	20	ℜ	ℜ	PROPN
ap-7678	305	21	dx	dx	PROPN
ap-7678	305	22	xpψ(x	xpψ(x	PROPN
ap-7678	305	23	)	)	PUNCT
ap-7678	305	24	.	.	PUNCT
ap-7678	306	1	upon	upon	SCONJ
ap-7678	306	2	multiplying	multiply	VERB
ap-7678	306	3	both	both	DET
ap-7678	306	4	sides	side	NOUN
ap-7678	306	5	of	of	ADP
ap-7678	306	6	eq	eq	PROPN
ap-7678	306	7	.	.	PUNCT
ap-7678	307	1	(	(	PUNCT
ap-7678	307	2	47	47	NUM
ap-7678	307	3	)	)	PUNCT
ap-7678	307	4	by	by	ADP
ap-7678	307	5	xp	xp	ADV
ap-7678	307	6	and	and	CCONJ
ap-7678	307	7	implementing	implement	VERB
ap-7678	307	8	an	an	DET
ap-7678	307	9	integration	integration	NOUN
ap-7678	307	10	by	by	ADP
ap-7678	307	11	parts	part	NOUN
ap-7678	307	12	analysis	analysis	NOUN
ap-7678	307	13	,	,	PUNCT
ap-7678	307	14	implicitly	implicitly	ADV
ap-7678	307	15	assuming	assume	VERB
ap-7678	307	16	that	that	SCONJ
ap-7678	307	17	one	one	NOUN
ap-7678	307	18	is	be	AUX
ap-7678	307	19	working	work	VERB
ap-7678	307	20	with	with	ADP
ap-7678	307	21	a	a	DET
ap-7678	307	22	discrete	discrete	ADJ
ap-7678	307	23	state	state	NOUN
ap-7678	307	24	,	,	PUNCT
ap-7678	307	25	69	69	NUM
ap-7678	307	26	carlos	carlos	PROPN
ap-7678	307	27	r.	r.	PROPN
ap-7678	307	28	handy	handy	PROPN
ap-7678	307	29	acta	acta	PROPN
ap-7678	307	30	polytechnica	polytechnica	PROPN
ap-7678	307	31	there	there	ADV
ap-7678	307	32	ensues	ensue	VERB
ap-7678	307	33	a	a	DET
ap-7678	307	34	hamburger	hamburger	NOUN
ap-7678	307	35	moment	moment	NOUN
ap-7678	307	36	equation	equation	NOUN
ap-7678	307	37	relation	relation	NOUN
ap-7678	307	38	(	(	PUNCT
ap-7678	307	39	mer	mer	NOUN
ap-7678	307	40	)	)	PUNCT
ap-7678	307	41	of	of	ADP
ap-7678	307	42	the	the	DET
ap-7678	307	43	form	form	NOUN
ap-7678	307	44	:	:	PUNCT
ap-7678	307	45	µ(p	µ(p	PROPN
ap-7678	307	46	+	+	PUNCT
ap-7678	308	1	6	6	NUM
ap-7678	308	2	)	)	PUNCT
ap-7678	308	3	=	=	SYM
ap-7678	308	4	−	−	NOUN
ap-7678	308	5	m	m	VERB
ap-7678	308	6	µ(p	µ(p	NOUN
ap-7678	308	7	+	+	PUNCT
ap-7678	308	8	2	2	X
ap-7678	308	9	)	)	PUNCT
ap-7678	308	10	+	+	CCONJ
ap-7678	308	11	eµ(p	eµ(p	NUM
ap-7678	308	12	)	)	PUNCT
ap-7678	309	1	+	+	CCONJ
ap-7678	309	2	p(p	p(p	ADV
ap-7678	309	3	−	−	NOUN
ap-7678	309	4	1)µ(p	1)µ(p	NUM
ap-7678	309	5	−	−	NOUN
ap-7678	309	6	2	2	NUM
ap-7678	309	7	)	)	PUNCT
ap-7678	309	8	,	,	PUNCT
ap-7678	309	9	p	p	PRON
ap-7678	309	10	≥	≥	NOUN
ap-7678	309	11	0	0	NUM
ap-7678	309	12	.	.	PUNCT
ap-7678	310	1	(	(	PUNCT
ap-7678	310	2	48	48	NUM
ap-7678	310	3	)	)	PUNCT
ap-7678	310	4	this	this	PRON
ap-7678	310	5	is	be	AUX
ap-7678	310	6	a	a	DET
ap-7678	310	7	mer	mer	NOUN
ap-7678	310	8	equation	equation	NOUN
ap-7678	310	9	of	of	ADP
ap-7678	310	10	order	order	NOUN
ap-7678	310	11	1	1	NUM
ap-7678	310	12	+	+	CCONJ
ap-7678	310	13	ms	ms	NOUN
ap-7678	310	14	=	=	PROPN
ap-7678	310	15	6	6	NUM
ap-7678	310	16	.	.	PUNCT
ap-7678	311	1	the	the	DET
ap-7678	311	2	mer	mer	PROPN
ap-7678	311	3	relation	relation	NOUN
ap-7678	311	4	in	in	ADP
ap-7678	311	5	eq	eq	ADP
ap-7678	311	6	.	.	PUNCT
ap-7678	312	1	(	(	PUNCT
ap-7678	312	2	48	48	NUM
ap-7678	312	3	)	)	PUNCT
ap-7678	312	4	applies	apply	VERB
ap-7678	312	5	to	to	ADP
ap-7678	312	6	both	both	CCONJ
ap-7678	312	7	even	even	ADV
ap-7678	312	8	and	and	CCONJ
ap-7678	312	9	odd	odd	ADJ
ap-7678	312	10	discrete	discrete	ADJ
ap-7678	312	11	states	state	NOUN
ap-7678	312	12	.	.	PUNCT
ap-7678	313	1	the	the	DET
ap-7678	313	2	generator	generator	NOUN
ap-7678	313	3	form	form	NOUN
ap-7678	313	4	for	for	ADP
ap-7678	313	5	the	the	DET
ap-7678	313	6	above	above	ADJ
ap-7678	313	7	mer	mer	NOUN
ap-7678	313	8	becomes	become	VERB
ap-7678	313	9	me(p	me(p	ADJ
ap-7678	313	10	+	+	CCONJ
ap-7678	313	11	6	6	NUM
ap-7678	313	12	,	,	PUNCT
ap-7678	313	13	ℓ	ℓ	NUM
ap-7678	313	14	)	)	PUNCT
ap-7678	313	15	=	=	SYM
ap-7678	314	1	−	−	PROPN
ap-7678	314	2	m	m	PRON
ap-7678	314	3	me(p	me(p	ADJ
ap-7678	314	4	+	+	CCONJ
ap-7678	314	5	2	2	NUM
ap-7678	314	6	,	,	PUNCT
ap-7678	314	7	ℓ	ℓ	NUM
ap-7678	314	8	)	)	PUNCT
ap-7678	314	9	+	+	CCONJ
ap-7678	314	10	eme(p	eme(p	PROPN
ap-7678	314	11	,	,	PUNCT
ap-7678	314	12	ℓ	ℓ	NOUN
ap-7678	314	13	)	)	PUNCT
ap-7678	315	1	+	+	CCONJ
ap-7678	315	2	p(p	p(p	ADV
ap-7678	315	3	−	−	NOUN
ap-7678	315	4	1)me(p	1)me(p	NUM
ap-7678	315	5	−	−	PROPN
ap-7678	315	6	2	2	NUM
ap-7678	315	7	,	,	PUNCT
ap-7678	315	8	ℓ	ℓ	NUM
ap-7678	315	9	)	)	PUNCT
ap-7678	315	10	,	,	PUNCT
ap-7678	315	11	(	(	PUNCT
ap-7678	315	12	49	49	NUM
ap-7678	315	13	)	)	PUNCT
ap-7678	315	14	p	p	X
ap-7678	315	15	≥	≥	NOUN
ap-7678	315	16	0	0	NUM
ap-7678	315	17	;	;	PUNCT
ap-7678	315	18	and	and	CCONJ
ap-7678	315	19	0	0	NUM
ap-7678	315	20	≤	≤	NUM
ap-7678	315	21	ℓ	ℓ	NOUN
ap-7678	315	22	≤	≤	NOUN
ap-7678	316	1	ms	ms	NOUN
ap-7678	316	2	=	=	SYM
ap-7678	316	3	5	5	NUM
ap-7678	316	4	.	.	PUNCT
ap-7678	317	1	the	the	DET
ap-7678	317	2	initialization	initialization	NOUN
ap-7678	317	3	conditions	condition	NOUN
ap-7678	317	4	becomes	become	VERB
ap-7678	317	5	m(ℓ1	m(ℓ1	NOUN
ap-7678	317	6	,	,	PUNCT
ap-7678	317	7	ℓ2	ℓ2	NOUN
ap-7678	317	8	)	)	PUNCT
ap-7678	317	9	=	=	SYM
ap-7678	317	10	δℓ1,ℓ2	δℓ1,ℓ2	ADJ
ap-7678	317	11	,	,	PUNCT
ap-7678	317	12	(	(	PUNCT
ap-7678	317	13	50	50	NUM
ap-7678	317	14	)	)	PUNCT
ap-7678	317	15	for	for	ADP
ap-7678	317	16	0	0	NUM
ap-7678	317	17	≤	≤	NOUN
ap-7678	317	18	ℓ1,2	ℓ1,2	ADJ
ap-7678	317	19	≤	≤	NOUN
ap-7678	317	20	5	5	NUM
ap-7678	317	21	.	.	PUNCT
ap-7678	318	1	the	the	DET
ap-7678	318	2	oppq	oppq	ADJ
ap-7678	318	3	representation	representation	NOUN
ap-7678	318	4	becomes	become	VERB
ap-7678	318	5	ψ(x	ψ(x	NOUN
ap-7678	318	6	)	)	PUNCT
ap-7678	319	1	=	=	PUNCT
ap-7678	320	1	∞∑	∞∑	PRON
ap-7678	320	2	n=0	n=0	NUM
ap-7678	320	3	cnpn(x)r(x	cnpn(x)r(x	NOUN
ap-7678	320	4	)	)	PUNCT
ap-7678	320	5	.	.	PUNCT
ap-7678	321	1	(	(	PUNCT
ap-7678	321	2	51	51	NUM
ap-7678	321	3	)	)	PUNCT
ap-7678	321	4	one	one	PRON
ap-7678	321	5	can	can	AUX
ap-7678	321	6	take	take	VERB
ap-7678	321	7	the	the	DET
ap-7678	321	8	weight	weight	NOUN
ap-7678	321	9	to	to	PART
ap-7678	321	10	be	be	AUX
ap-7678	321	11	the	the	DET
ap-7678	321	12	gaussian	gaussian	NOUN
ap-7678	321	13	,	,	PUNCT
ap-7678	321	14	rg(x	rg(x	NUM
ap-7678	321	15	)	)	PUNCT
ap-7678	321	16	=	=	PRON
ap-7678	321	17	exp(−x2	exp(−x2	NOUN
ap-7678	321	18	)	)	PUNCT
ap-7678	321	19	,	,	PUNCT
ap-7678	321	20	or	or	CCONJ
ap-7678	321	21	the	the	DET
ap-7678	321	22	asymptotic	asymptotic	ADJ
ap-7678	321	23	form	form	NOUN
ap-7678	321	24	for	for	ADP
ap-7678	321	25	the	the	DET
ap-7678	321	26	physical	physical	ADJ
ap-7678	321	27	states	state	NOUN
ap-7678	321	28	,	,	PUNCT
ap-7678	321	29	ra(x	ra(x	NOUN
ap-7678	321	30	)	)	PUNCT
ap-7678	321	31	=	=	X
ap-7678	321	32	exp(−	exp(−	PROPN
ap-7678	321	33	x4	x4	PROPN
ap-7678	321	34	4	4	NUM
ap-7678	321	35	)	)	PUNCT
ap-7678	321	36	.	.	PUNCT
ap-7678	322	1	in	in	ADP
ap-7678	322	2	the	the	DET
ap-7678	322	3	original	original	ADJ
ap-7678	322	4	formulation	formulation	NOUN
ap-7678	322	5	of	of	ADP
ap-7678	322	6	the	the	DET
ap-7678	322	7	oppq	oppq	NOUN
ap-7678	322	8	-	-	PUNCT
ap-7678	322	9	approximation	approximation	NOUN
ap-7678	322	10	method	method	NOUN
ap-7678	322	11	(	(	PUNCT
ap-7678	322	12	oppq	oppq	PROPN
ap-7678	322	13	-	-	PUNCT
ap-7678	322	14	am	am	NOUN
ap-7678	322	15	)	)	PUNCT
ap-7678	322	16	handy	handy	ADJ
ap-7678	322	17	and	and	CCONJ
ap-7678	322	18	vrinceanu	vrinceanu	NOUN
ap-7678	322	19	examined	examine	VERB
ap-7678	322	20	both	both	PRON
ap-7678	322	21	;	;	PUNCT
ap-7678	322	22	and	and	CCONJ
ap-7678	322	23	demonstrated	demonstrate	VERB
ap-7678	322	24	the	the	DET
ap-7678	322	25	superiority	superiority	NOUN
ap-7678	322	26	of	of	ADP
ap-7678	322	27	ra(x	ra(x	NOUN
ap-7678	322	28	)	)	PUNCT
ap-7678	322	29	.	.	PUNCT
ap-7678	323	1	our	our	PRON
ap-7678	323	2	interest	interest	NOUN
ap-7678	323	3	here	here	ADV
ap-7678	323	4	is	be	AUX
ap-7678	323	5	to	to	PART
ap-7678	323	6	demonstrate	demonstrate	VERB
ap-7678	323	7	this	this	DET
ap-7678	323	8	approach	approach	NOUN
ap-7678	323	9	,	,	PUNCT
ap-7678	323	10	and	and	CCONJ
ap-7678	323	11	to	to	PART
ap-7678	323	12	implement	implement	VERB
ap-7678	323	13	the	the	DET
ap-7678	323	14	oppq	oppq	NOUN
ap-7678	323	15	-	-	PUNCT
ap-7678	323	16	bounding	bound	VERB
ap-7678	323	17	method	method	NOUN
ap-7678	323	18	with	with	ADP
ap-7678	323	19	regards	regard	NOUN
ap-7678	323	20	to	to	ADP
ap-7678	323	21	the	the	DET
ap-7678	323	22	ra	ra	PROPN
ap-7678	323	23	formulation	formulation	NOUN
ap-7678	323	24	.	.	PUNCT
ap-7678	324	1	the	the	DET
ap-7678	324	2	orthonormal	orthonormal	ADJ
ap-7678	324	3	polynomials	polynomial	NOUN
ap-7678	324	4	for	for	ADP
ap-7678	324	5	the	the	DET
ap-7678	324	6	gaussian	gaussian	NOUN
ap-7678	324	7	,	,	PUNCT
ap-7678	324	8	rg(x	rg(x	NUM
ap-7678	324	9	)	)	PUNCT
ap-7678	324	10	,	,	PUNCT
ap-7678	324	11	are	be	AUX
ap-7678	324	12	determined	determine	VERB
ap-7678	324	13	by	by	ADP
ap-7678	324	14	the	the	DET
ap-7678	324	15	hermite	hermite	ADJ
ap-7678	324	16	polynomials	polynomial	VERB
ap-7678	324	17	⟨ĥm|exp(−x2)|ĥn⟩	⟨ĥm|exp(−x2)|ĥn⟩	PROPN
ap-7678	325	1	=	=	PUNCT
ap-7678	325	2	δm	δm	PROPN
ap-7678	325	3	,	,	PUNCT
ap-7678	325	4	n	n	CCONJ
ap-7678	325	5	,	,	PUNCT
ap-7678	325	6	(	(	PUNCT
ap-7678	325	7	52	52	NUM
ap-7678	325	8	)	)	PUNCT
ap-7678	325	9	or	or	CCONJ
ap-7678	325	10	ĥn(x	ĥn(x	X
ap-7678	325	11	)	)	PUNCT
ap-7678	325	12	=	=	SYM
ap-7678	326	1	1√√	1√√	NUM
ap-7678	326	2	π2nn	π2nn	NOUN
ap-7678	326	3	!	!	PUNCT
ap-7678	326	4	hn(x	hn(x	X
ap-7678	326	5	)	)	PUNCT
ap-7678	326	6	.	.	PUNCT
ap-7678	327	1	(	(	PUNCT
ap-7678	327	2	53	53	NUM
ap-7678	327	3	)	)	PUNCT
ap-7678	327	4	the	the	DET
ap-7678	327	5	orthonormal	orthonormal	ADJ
ap-7678	327	6	polynomials	polynomial	NOUN
ap-7678	327	7	(	(	PUNCT
ap-7678	327	8	i.e.	i.e.	X
ap-7678	327	9	their	their	PRON
ap-7678	327	10	coefficients	coefficient	NOUN
ap-7678	327	11	)	)	PUNCT
ap-7678	327	12	for	for	ADP
ap-7678	327	13	ra(x	ra(x	NOUN
ap-7678	327	14	)	)	PUNCT
ap-7678	327	15	are	be	AUX
ap-7678	327	16	determined	determine	VERB
ap-7678	327	17	through	through	ADP
ap-7678	327	18	the	the	DET
ap-7678	327	19	cholesky	cholesky	ADJ
ap-7678	327	20	decomposition	decomposition	NOUN
ap-7678	327	21	of	of	ADP
ap-7678	327	22	the	the	DET
ap-7678	327	23	hankel	hankel	NOUN
ap-7678	327	24	moment	moment	NOUN
ap-7678	327	25	matrix	matrix	NOUN
ap-7678	327	26	wm	wm	PROPN
ap-7678	327	27	,	,	PUNCT
ap-7678	327	28	n	n	PROPN
ap-7678	327	29	≡	≡	PROPN
ap-7678	327	30	∫	∫	PROPN
ap-7678	327	31	ℜ	ℜ	PROPN
ap-7678	327	32	dx	dx	PROPN
ap-7678	327	33	xm+nexp(−x4/4	xm+nexp(−x4/4	PROPN
ap-7678	327	34	)	)	PUNCT
ap-7678	327	35	,	,	PUNCT
ap-7678	327	36	(	(	PUNCT
ap-7678	327	37	54	54	NUM
ap-7678	327	38	)	)	PUNCT
ap-7678	327	39	or	or	CCONJ
ap-7678	327	40	wm	wm	PROPN
ap-7678	327	41	,	,	PUNCT
ap-7678	327	42	n	n	NOUN
ap-7678	327	43	=	=	PUNCT
ap-7678	327	44	{	{	PUNCT
ap-7678	327	45	0	0	NUM
ap-7678	327	46	,	,	PUNCT
ap-7678	327	47	if	if	SCONJ
ap-7678	327	48	m	m	VERB
ap-7678	327	49	+	+	NOUN
ap-7678	327	50	n	n	CCONJ
ap-7678	327	51	=	=	SYM
ap-7678	327	52	odd	odd	ADJ
ap-7678	327	53	,	,	PUNCT
ap-7678	327	54	2η−	2η−	PROPN
ap-7678	327	55	1	1	NUM
ap-7678	327	56	2	2	NUM
ap-7678	327	57	γ	γ	X
ap-7678	327	58	(	(	PUNCT
ap-7678	327	59	η+1/2	η+1/2	NOUN
ap-7678	327	60	2	2	NUM
ap-7678	327	61	)	)	PUNCT
ap-7678	327	62	,	,	PUNCT
ap-7678	327	63	if	if	SCONJ
ap-7678	327	64	m	m	VERB
ap-7678	327	65	+	+	NUM
ap-7678	327	66	n	n	PROPN
ap-7678	327	67	=	=	SYM
ap-7678	327	68	2η	2η	PROPN
ap-7678	327	69	,	,	PUNCT
ap-7678	327	70	(	(	PUNCT
ap-7678	327	71	even	even	ADV
ap-7678	327	72	)	)	PUNCT
ap-7678	327	73	.	.	PUNCT
ap-7678	328	1	(	(	PUNCT
ap-7678	328	2	55	55	NUM
ap-7678	328	3	)	)	PUNCT
ap-7678	328	4	the	the	DET
ap-7678	328	5	projection	projection	NOUN
ap-7678	328	6	coefficients	coefficient	NOUN
ap-7678	328	7	are	be	AUX
ap-7678	328	8	determined	determine	VERB
ap-7678	328	9	by	by	ADP
ap-7678	328	10	cn(e	cn(e	PROPN
ap-7678	328	11	,	,	PUNCT
ap-7678	328	12	µ0	µ0	PROPN
ap-7678	328	13	,	,	PUNCT
ap-7678	328	14	.	.	PUNCT
ap-7678	328	15	.	.	PUNCT
ap-7678	329	1	.	.	PUNCT
ap-7678	330	1	,	,	PUNCT
ap-7678	330	2	µ5	µ5	PROPN
ap-7678	330	3	)	)	PUNCT
ap-7678	331	1	=	=	SYM
ap-7678	331	2	ms=5∑	ms=5∑	NOUN
ap-7678	331	3	ℓ=0	ℓ=0	SYM
ap-7678	331	4	λ(n	λ(n	PROPN
ap-7678	331	5	)	)	PUNCT
ap-7678	331	6	ℓ	ℓ	PROPN
ap-7678	331	7	(	(	PUNCT
ap-7678	331	8	e	e	NOUN
ap-7678	331	9	)	)	PUNCT
ap-7678	331	10	µℓ	µℓ	ADP
ap-7678	331	11	,	,	PUNCT
ap-7678	331	12	(	(	PUNCT
ap-7678	331	13	56	56	NUM
ap-7678	331	14	)	)	PUNCT
ap-7678	331	15	from	from	ADP
ap-7678	331	16	eq	eq	ADP
ap-7678	331	17	.	.	PUNCT
ap-7678	332	1	(	(	PUNCT
ap-7678	332	2	17	17	NUM
ap-7678	332	3	-	-	SYM
ap-7678	332	4	18	18	NUM
ap-7678	332	5	)	)	PUNCT
ap-7678	332	6	,	,	PUNCT
ap-7678	332	7	where	where	SCONJ
ap-7678	332	8	λ(n	λ(n	PROPN
ap-7678	332	9	)	)	PUNCT
ap-7678	332	10	ℓ	ℓ	PROPN
ap-7678	332	11	(	(	PUNCT
ap-7678	332	12	e	e	NOUN
ap-7678	332	13	)	)	PUNCT
ap-7678	332	14	=	=	SYM
ap-7678	332	15	n∑	n∑	NOUN
ap-7678	332	16	j=0	j=0	PROPN
ap-7678	332	17	ξ(n	ξ(n	PROPN
ap-7678	332	18	)	)	PUNCT
ap-7678	332	19	j	j	PROPN
ap-7678	332	20	me(j	me(j	NOUN
ap-7678	332	21	,	,	PUNCT
ap-7678	332	22	ℓ	ℓ	NOUN
ap-7678	332	23	)	)	PUNCT
ap-7678	332	24	.	.	PUNCT
ap-7678	333	1	(	(	PUNCT
ap-7678	333	2	57	57	NUM
ap-7678	333	3	)	)	PUNCT
ap-7678	333	4	the	the	DET
ap-7678	333	5	oppq	oppq	NOUN
ap-7678	333	6	-	-	PUNCT
ap-7678	333	7	approximation	approximation	NOUN
ap-7678	333	8	method	method	NOUN
ap-7678	333	9	corresponds	correspond	VERB
ap-7678	333	10	to	to	ADP
ap-7678	333	11	solving	solve	VERB
ap-7678	333	12	the	the	DET
ap-7678	333	13	secular	secular	ADJ
ap-7678	333	14	equation	equation	NOUN
ap-7678	333	15	cn−ℓ1(e	cn−ℓ1(e	NOUN
ap-7678	333	16	,	,	PUNCT
ap-7678	333	17	−→µ	−→µ	NOUN
ap-7678	333	18	)	)	PUNCT
ap-7678	333	19	=	=	SYM
ap-7678	333	20	0	0	NUM
ap-7678	333	21	,	,	PUNCT
ap-7678	333	22	(	(	PUNCT
ap-7678	333	23	58	58	NUM
ap-7678	333	24	)	)	PUNCT
ap-7678	333	25	for	for	ADP
ap-7678	333	26	0	0	NUM
ap-7678	333	27	≤	≤	NUM
ap-7678	333	28	ℓ	ℓ	NOUN
ap-7678	333	29	≤	≤	NUM
ap-7678	333	30	5	5	NUM
ap-7678	333	31	and	and	CCONJ
ap-7678	333	32	n	n	NOUN
ap-7678	333	33	→	→	SYM
ap-7678	333	34	∞	∞	PROPN
ap-7678	333	35	,	,	PUNCT
ap-7678	333	36	or	or	CCONJ
ap-7678	333	37	det	det	PROPN
ap-7678	333	38	(	(	PUNCT
ap-7678	333	39	λ(n−ℓ1	λ(n−ℓ1	NOUN
ap-7678	333	40	)	)	PUNCT
ap-7678	333	41	ℓ2	ℓ2	NOUN
ap-7678	333	42	(	(	PUNCT
ap-7678	333	43	e	e	NOUN
ap-7678	333	44	)	)	PUNCT
ap-7678	333	45	)	)	PUNCT
ap-7678	334	1	=	=	SYM
ap-7678	334	2	0	0	NUM
ap-7678	334	3	,	,	PUNCT
ap-7678	334	4	(	(	PUNCT
ap-7678	334	5	59	59	NUM
ap-7678	334	6	)	)	PUNCT
ap-7678	334	7	a	a	DET
ap-7678	334	8	6	6	NUM
ap-7678	334	9	×	×	NOUN
ap-7678	334	10	6	6	NUM
ap-7678	334	11	determinantal	determinantal	ADJ
ap-7678	334	12	secular	secular	ADJ
ap-7678	334	13	equation	equation	NOUN
ap-7678	334	14	.	.	PUNCT
ap-7678	335	1	in	in	ADP
ap-7678	335	2	summary	summary	NOUN
ap-7678	335	3	,	,	PUNCT
ap-7678	335	4	having	having	AUX
ap-7678	335	5	chosen	choose	VERB
ap-7678	335	6	n	n	PROPN
ap-7678	335	7	,	,	PUNCT
ap-7678	335	8	we	we	PRON
ap-7678	335	9	need	need	VERB
ap-7678	335	10	to	to	PART
ap-7678	335	11	generate	generate	VERB
ap-7678	335	12	{	{	PUNCT
ap-7678	335	13	λ(n	λ(n	PROPN
ap-7678	335	14	)	)	PUNCT
ap-7678	335	15	ℓ	ℓ	PROPN
ap-7678	335	16	(	(	PUNCT
ap-7678	335	17	e)|n	e)|n	X
ap-7678	335	18	−	−	PROPN
ap-7678	335	19	ms	ms	PROPN
ap-7678	335	20	≤	≤	PUNCT
ap-7678	335	21	n	n	CCONJ
ap-7678	335	22	≤	≤	NOUN
ap-7678	335	23	n	n	CCONJ
ap-7678	335	24	}	}	PUNCT
ap-7678	335	25	.	.	PUNCT
ap-7678	336	1	this	this	PRON
ap-7678	336	2	requires	require	VERB
ap-7678	336	3	the	the	DET
ap-7678	336	4	orthonormal	orthonormal	ADJ
ap-7678	336	5	polynomials	polynomial	NOUN
ap-7678	336	6	to	to	PART
ap-7678	336	7	order	order	VERB
ap-7678	336	8	n	n	PRON
ap-7678	336	9	and	and	CCONJ
ap-7678	336	10	the	the	DET
ap-7678	336	11	generation	generation	NOUN
ap-7678	336	12	of	of	ADP
ap-7678	336	13	me(p	me(p	NOUN
ap-7678	336	14	+	+	CCONJ
ap-7678	336	15	6	6	NUM
ap-7678	336	16	,	,	PUNCT
ap-7678	336	17	ℓ	ℓ	NUM
ap-7678	336	18	)	)	PUNCT
ap-7678	336	19	,	,	PUNCT
ap-7678	336	20	for	for	ADP
ap-7678	336	21	0	0	NUM
ap-7678	336	22	≤	≤	NOUN
ap-7678	336	23	p	p	NOUN
ap-7678	336	24	≤	≤	NOUN
ap-7678	336	25	n	n	CCONJ
ap-7678	336	26	−	−	PROPN
ap-7678	336	27	6	6	NUM
ap-7678	336	28	.	.	PUNCT
ap-7678	337	1	the	the	DET
ap-7678	337	2	hankel	hankel	NOUN
ap-7678	337	3	moment	moment	NOUN
ap-7678	337	4	matrix	matrix	NOUN
ap-7678	337	5	for	for	ADP
ap-7678	337	6	the	the	DET
ap-7678	337	7	weight	weight	NOUN
ap-7678	337	8	are	be	AUX
ap-7678	337	9	required	require	VERB
ap-7678	337	10	up	up	ADP
ap-7678	337	11	to	to	PART
ap-7678	337	12	order	order	VERB
ap-7678	337	13	2n	2n	NUM
ap-7678	337	14	.	.	PUNCT
ap-7678	338	1	the	the	DET
ap-7678	338	2	results	result	NOUN
ap-7678	338	3	for	for	ADP
ap-7678	338	4	both	both	DET
ap-7678	338	5	choices	choice	NOUN
ap-7678	338	6	of	of	ADP
ap-7678	338	7	weight	weight	NOUN
ap-7678	338	8	are	be	AUX
ap-7678	338	9	indicated	indicate	VERB
ap-7678	338	10	in	in	ADP
ap-7678	338	11	tables	table	NOUN
ap-7678	338	12	1	1	NUM
ap-7678	338	13	and	and	CCONJ
ap-7678	338	14	2	2	NUM
ap-7678	338	15	,	,	PUNCT
ap-7678	338	16	for	for	ADP
ap-7678	338	17	the	the	DET
ap-7678	338	18	first	first	ADJ
ap-7678	338	19	ten	ten	NUM
ap-7678	338	20	discrete	discrete	ADJ
ap-7678	338	21	state	state	NOUN
ap-7678	338	22	energies	energy	NOUN
ap-7678	338	23	.	.	PUNCT
ap-7678	339	1	it	it	PRON
ap-7678	339	2	is	be	AUX
ap-7678	339	3	clear	clear	ADJ
ap-7678	339	4	that	that	SCONJ
ap-7678	339	5	the	the	DET
ap-7678	339	6	ra	ra	PROPN
ap-7678	339	7	choice	choice	NOUN
ap-7678	339	8	for	for	ADP
ap-7678	339	9	the	the	DET
ap-7678	339	10	weight	weight	NOUN
ap-7678	339	11	is	be	AUX
ap-7678	339	12	orders	order	NOUN
ap-7678	339	13	of	of	ADP
ap-7678	339	14	magnitude	magnitude	NOUN
ap-7678	339	15	faster	fast	ADV
ap-7678	339	16	than	than	ADP
ap-7678	339	17	the	the	DET
ap-7678	339	18	simple	simple	ADJ
ap-7678	339	19	gaussian	gaussian	NOUN
ap-7678	339	20	.	.	PUNCT
ap-7678	340	1	it	it	PRON
ap-7678	340	2	is	be	AUX
ap-7678	340	3	natural	natural	ADJ
ap-7678	340	4	to	to	PART
ap-7678	340	5	normalize	normalize	VERB
ap-7678	340	6	the	the	DET
ap-7678	340	7	missing	miss	VERB
ap-7678	340	8	moments	moment	NOUN
ap-7678	340	9	according	accord	VERB
ap-7678	340	10	to	to	ADP
ap-7678	340	11	a	a	DET
ap-7678	340	12	unit	unit	NOUN
ap-7678	340	13	normalization	normalization	NOUN
ap-7678	340	14	|−→µ	|−→µ	VERB
ap-7678	340	15	|2	|2	NUM
ap-7678	340	16	=	=	SYM
ap-7678	340	17	1	1	NUM
ap-7678	340	18	,	,	PUNCT
ap-7678	340	19	or	or	CCONJ
ap-7678	340	20	5∑	5∑	PROPN
ap-7678	340	21	ℓ=0	ℓ=0	SYM
ap-7678	341	1	µ2	µ2	PROPN
ap-7678	341	2	ℓ	ℓ	PROPN
ap-7678	341	3	=	=	SYM
ap-7678	341	4	1	1	X
ap-7678	341	5	.	.	PUNCT
ap-7678	341	6	(	(	PUNCT
ap-7678	341	7	60	60	NUM
ap-7678	341	8	)	)	PUNCT
ap-7678	341	9	accordingly	accordingly	ADV
ap-7678	341	10	,	,	PUNCT
ap-7678	341	11	the	the	DET
ap-7678	341	12	energy	energy	NOUN
ap-7678	341	13	functional	functional	NOUN
ap-7678	341	14	whose	whose	DET
ap-7678	341	15	minimization	minimization	NOUN
ap-7678	341	16	is	be	AUX
ap-7678	341	17	part	part	NOUN
ap-7678	341	18	of	of	ADP
ap-7678	341	19	the	the	DET
ap-7678	341	20	oppq	oppq	NOUN
ap-7678	341	21	-	-	PUNCT
ap-7678	341	22	bounding	bound	VERB
ap-7678	341	23	method	method	NOUN
ap-7678	341	24	becomes	become	VERB
ap-7678	341	25	:	:	PUNCT
ap-7678	341	26	ln	ln	ADJ
ap-7678	341	27	(	(	PUNCT
ap-7678	341	28	e	e	NOUN
ap-7678	341	29	)	)	PUNCT
ap-7678	341	30	→	→	SYM
ap-7678	341	31	λn	λn	NOUN
ap-7678	341	32	(	(	PUNCT
ap-7678	341	33	e	e	NOUN
ap-7678	341	34	)	)	PUNCT
ap-7678	341	35	≡	≡	PROPN
ap-7678	341	36	smallest	small	ADJ
ap-7678	341	37	eigenvalue	eigenvalue	NOUN
ap-7678	341	38	of	of	ADP
ap-7678	341	39	pn	pn	PROPN
ap-7678	341	40	(	(	PUNCT
ap-7678	341	41	e	e	NOUN
ap-7678	341	42	)	)	PUNCT
ap-7678	341	43	,	,	PUNCT
ap-7678	341	44	(	(	PUNCT
ap-7678	341	45	61	61	NUM
ap-7678	341	46	)	)	PUNCT
ap-7678	341	47	where	where	SCONJ
ap-7678	341	48	the	the	DET
ap-7678	341	49	dyad	dyad	NOUN
ap-7678	341	50	matrix	matrix	NOUN
ap-7678	341	51	is	be	AUX
ap-7678	341	52	given	give	VERB
ap-7678	341	53	by	by	ADP
ap-7678	341	54	(	(	PUNCT
ap-7678	341	55	pn	pn	PROPN
ap-7678	341	56	(	(	PUNCT
ap-7678	341	57	e	e	NOUN
ap-7678	341	58	)	)	PUNCT
ap-7678	341	59	)	)	PUNCT
ap-7678	342	1	ℓ1,ℓ2	ℓ1,ℓ2	PROPN
ap-7678	342	2	=	=	SYM
ap-7678	342	3	n∑	n∑	PROPN
ap-7678	342	4	j=0	j=0	PROPN
ap-7678	342	5	λ(j	λ(j	PROPN
ap-7678	342	6	)	)	PUNCT
ap-7678	342	7	ℓ1	ℓ1	NOUN
ap-7678	342	8	(	(	PUNCT
ap-7678	342	9	e)λ(j	e)λ(j	NOUN
ap-7678	342	10	)	)	PUNCT
ap-7678	342	11	ℓ2	ℓ2	NOUN
ap-7678	342	12	(	(	PUNCT
ap-7678	342	13	e	e	NOUN
ap-7678	342	14	)	)	PUNCT
ap-7678	342	15	.	.	PUNCT
ap-7678	343	1	(	(	PUNCT
ap-7678	343	2	62	62	NUM
ap-7678	343	3	)	)	PUNCT
ap-7678	343	4	the	the	DET
ap-7678	343	5	λn	λn	NOUN
ap-7678	343	6	(	(	PUNCT
ap-7678	343	7	e	e	NOUN
ap-7678	343	8	)	)	PUNCT
ap-7678	343	9	form	form	VERB
ap-7678	343	10	a	a	DET
ap-7678	343	11	family	family	NOUN
ap-7678	343	12	of	of	ADP
ap-7678	343	13	nested	nested	ADJ
ap-7678	343	14	,	,	PUNCT
ap-7678	343	15	increasing	increase	VERB
ap-7678	343	16	functions	function	NOUN
ap-7678	343	17	,	,	PUNCT
ap-7678	343	18	whose	whose	DET
ap-7678	343	19	local	local	ADJ
ap-7678	343	20	minima	minima	NOUN
ap-7678	343	21	approximate	approximate	NOUN
ap-7678	343	22	the	the	DET
ap-7678	343	23	eigenenergies	eigenenergie	NOUN
ap-7678	343	24	,	,	PUNCT
ap-7678	343	25	and	and	CCONJ
ap-7678	343	26	serve	serve	VERB
ap-7678	343	27	to	to	PART
ap-7678	343	28	define	define	VERB
ap-7678	343	29	a	a	DET
ap-7678	343	30	bounding	bounding	NOUN
ap-7678	343	31	formalism	formalism	NOUN
ap-7678	343	32	.	.	PUNCT
ap-7678	344	1	we	we	PRON
ap-7678	344	2	depict	depict	VERB
ap-7678	344	3	this	this	DET
ap-7678	344	4	behavior	behavior	NOUN
ap-7678	344	5	for	for	ADP
ap-7678	344	6	the	the	DET
ap-7678	344	7	ra(x	ra(x	NOUN
ap-7678	344	8	)	)	PUNCT
ap-7678	344	9	weight	weight	NOUN
ap-7678	344	10	,	,	PUNCT
ap-7678	344	11	in	in	ADP
ap-7678	344	12	figure	figure	NOUN
ap-7678	344	13	1	1	NUM
ap-7678	344	14	.	.	PUNCT
ap-7678	345	1	the	the	DET
ap-7678	345	2	resolution	resolution	NOUN
ap-7678	345	3	is	be	AUX
ap-7678	345	4	not	not	PART
ap-7678	345	5	too	too	ADV
ap-7678	345	6	high	high	ADJ
ap-7678	345	7	and	and	CCONJ
ap-7678	345	8	so	so	ADV
ap-7678	345	9	it	it	PRON
ap-7678	345	10	is	be	AUX
ap-7678	345	11	difficult	difficult	ADJ
ap-7678	345	12	to	to	PART
ap-7678	345	13	appreciate	appreciate	VERB
ap-7678	345	14	that	that	SCONJ
ap-7678	345	15	the	the	DET
ap-7678	345	16	downward	downward	ADJ
ap-7678	345	17	spikes	spike	NOUN
ap-7678	345	18	actually	actually	ADV
ap-7678	345	19	are	be	AUX
ap-7678	345	20	very	very	ADV
ap-7678	345	21	close	close	ADJ
ap-7678	345	22	to	to	ADP
ap-7678	345	23	each	each	DET
ap-7678	345	24	other	other	ADJ
ap-7678	345	25	.	.	PUNCT
ap-7678	346	1	this	this	DET
ap-7678	346	2	type	type	NOUN
ap-7678	346	3	of	of	ADP
ap-7678	346	4	illustration	illustration	NOUN
ap-7678	346	5	becomes	become	VERB
ap-7678	346	6	easier	easy	ADJ
ap-7678	346	7	to	to	PART
ap-7678	346	8	recognize	recognize	VERB
ap-7678	346	9	in	in	ADP
ap-7678	346	10	a	a	DET
ap-7678	346	11	subsequent	subsequent	ADJ
ap-7678	346	12	reformulation	reformulation	NOUN
ap-7678	346	13	of	of	ADP
ap-7678	346	14	the	the	DET
ap-7678	346	15	sextic	sextic	ADJ
ap-7678	346	16	anharmonic	anharmonic	ADJ
ap-7678	346	17	oscillator	oscillator	NOUN
ap-7678	346	18	.	.	PUNCT
ap-7678	347	1	we	we	PRON
ap-7678	347	2	can	can	AUX
ap-7678	347	3	also	also	ADV
ap-7678	347	4	apply	apply	VERB
ap-7678	347	5	emm	emm	PROPN
ap-7678	347	6	analysis	analysis	NOUN
ap-7678	347	7	to	to	ADP
ap-7678	347	8	the	the	DET
ap-7678	347	9	system	system	NOUN
ap-7678	347	10	in	in	ADP
ap-7678	347	11	eq	eq	ADP
ap-7678	347	12	.	.	PUNCT
ap-7678	348	1	(	(	PUNCT
ap-7678	348	2	47	47	NUM
ap-7678	348	3	)	)	PUNCT
ap-7678	348	4	.	.	PUNCT
ap-7678	349	1	the	the	DET
ap-7678	349	2	emm	emm	PROPN
ap-7678	349	3	procedure	procedure	NOUN
ap-7678	349	4	essentially	essentially	ADV
ap-7678	349	5	imposes	impose	VERB
ap-7678	349	6	the	the	DET
ap-7678	349	7	well	well	ADV
ap-7678	349	8	known	know	VERB
ap-7678	349	9	hankel	hankel	NOUN
ap-7678	349	10	hadamard	hadamard	ADJ
ap-7678	349	11	moment	moment	NOUN
ap-7678	349	12	problem	problem	NOUN
ap-7678	349	13	constraints	constraint	NOUN
ap-7678	349	14	in	in	ADP
ap-7678	349	15	order	order	NOUN
ap-7678	349	16	to	to	PART
ap-7678	349	17	bound	bound	VERB
ap-7678	349	18	the	the	DET
ap-7678	349	19	discrete	discrete	ADJ
ap-7678	349	20	state	state	NOUN
ap-7678	349	21	energies	energy	NOUN
ap-7678	349	22	associated	associate	VERB
ap-7678	349	23	with	with	ADP
ap-7678	349	24	nonnegative	nonnegative	ADJ
ap-7678	349	25	configuration	configuration	NOUN
ap-7678	349	26	space	space	NOUN
ap-7678	349	27	solutions	solution	NOUN
ap-7678	349	28	.	.	PUNCT
ap-7678	350	1	since	since	SCONJ
ap-7678	350	2	the	the	DET
ap-7678	350	3	only	only	ADJ
ap-7678	350	4	state	state	NOUN
ap-7678	350	5	of	of	ADP
ap-7678	350	6	this	this	DET
ap-7678	350	7	type	type	NOUN
ap-7678	350	8	is	be	AUX
ap-7678	350	9	the	the	DET
ap-7678	350	10	ground	ground	NOUN
ap-7678	350	11	state	state	NOUN
ap-7678	350	12	,	,	PUNCT
ap-7678	350	13	which	which	PRON
ap-7678	350	14	must	must	AUX
ap-7678	350	15	also	also	ADV
ap-7678	350	16	be	be	AUX
ap-7678	350	17	of	of	ADP
ap-7678	350	18	even	even	ADJ
ap-7678	350	19	parity	parity	NOUN
ap-7678	350	20	,	,	PUNCT
ap-7678	350	21	we	we	PRON
ap-7678	350	22	can	can	AUX
ap-7678	350	23	further	far	ADV
ap-7678	350	24	specialize	specialize	VERB
ap-7678	350	25	the	the	DET
ap-7678	350	26	mer	mer	PROPN
ap-7678	350	27	relation	relation	NOUN
ap-7678	350	28	in	in	ADP
ap-7678	350	29	eq	eq	ADP
ap-7678	350	30	.	.	PUNCT
ap-7678	351	1	(	(	PUNCT
ap-7678	351	2	48	48	NUM
ap-7678	351	3	)	)	PUNCT
ap-7678	351	4	to	to	ADP
ap-7678	351	5	70	70	NUM
ap-7678	351	6	vol	vol	NOUN
ap-7678	351	7	.	.	PUNCT
ap-7678	352	1	62	62	NUM
ap-7678	352	2	no	no	INTJ
ap-7678	352	3	.	.	PUNCT
ap-7678	353	1	1/2022	1/2022	NUM
ap-7678	353	2	algebraic	algebraic	ADJ
ap-7678	353	3	eigenenergy	eigenenergy	NOUN
ap-7678	353	4	bounding	bounding	NOUN
ap-7678	353	5	method	method	NOUN
ap-7678	353	6	n	n	PRON
ap-7678	353	7	e0	e0	PROPN
ap-7678	353	8	e1	e1	PROPN
ap-7678	353	9	e2	e2	PROPN
ap-7678	353	10	e3	e3	PROPN
ap-7678	353	11	e4	e4	PROPN
ap-7678	353	12	25	25	NUM
ap-7678	353	13	-0.530376450630854	-0.530376450630854	NOUN
ap-7678	353	14	0.985067365669966	0.985067365669966	NUM
ap-7678	353	15	5.28830977093027	5.28830977093027	NUM
ap-7678	353	16	10.4934190522107	10.4934190522107	NUM
ap-7678	353	17	15.8338485774860	15.8338485774860	NUM
ap-7678	353	18	50	50	NUM
ap-7678	353	19	-0.523284216533135	-0.523284216533135	NOUN
ap-7678	353	20	1.00560182885171	1.00560182885171	NUM
ap-7678	353	21	5.37480811122565	5.37480811122565	NUM
ap-7678	353	22	10.5699924952422	10.5699924952422	NUM
ap-7678	353	23	16.7909114406834	16.7909114406834	NUM
ap-7678	353	24	75	75	NUM
ap-7678	353	25	-0.523268805558542	-0.523268805558542	NOUN
ap-7678	353	26	1.00576819439460	1.00576819439460	NUM
ap-7678	353	27	5.37496683348150	5.37496683348150	NUM
ap-7678	353	28	10.5725844031665	10.5725844031665	NUM
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ap-7678	353	35	16.7953468181267	16.7953468181267	NUM
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ap-7678	353	59	16.7953468327042	16.7953468327042	NUM
ap-7678	353	60	25	25	NUM
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ap-7678	353	74	1.00576834022554	1.00576834022554	NUM
ap-7678	353	75	5.37497000884004	5.37497000884004	NUM
ap-7678	353	76	10.5725850445859	10.5725850445859	NUM
ap-7678	353	77	16.7953468327036	16.7953468327036	NUM
ap-7678	353	78	200	200	NUM
ap-7678	353	79	.	.	PUNCT
ap-7678	353	80	-0.523268622127552	-0.523268622127552	NOUN
ap-7678	354	1	1.00576834022554	1.00576834022554	NUM
ap-7678	354	2	5.37497000884004	5.37497000884004	NUM
ap-7678	354	3	10.5725850445859	10.5725850445859	NUM
ap-7678	354	4	16.7953468327036	16.7953468327036	NUM
ap-7678	354	5	table	table	NOUN
ap-7678	354	6	1	1	NUM
ap-7678	354	7	.	.	PUNCT
ap-7678	355	1	oppq	oppq	PROPN
ap-7678	355	2	-	-	PUNCT
ap-7678	355	3	am	am	PROPN
ap-7678	355	4	,	,	PUNCT
ap-7678	355	5	for	for	ADP
ap-7678	355	6	the	the	DET
ap-7678	355	7	first	first	ADJ
ap-7678	355	8	five	five	NUM
ap-7678	355	9	states	state	NOUN
ap-7678	355	10	of	of	ADP
ap-7678	355	11	v	v	NOUN
ap-7678	355	12	(	(	PUNCT
ap-7678	355	13	x	x	NOUN
ap-7678	355	14	)	)	PUNCT
ap-7678	355	15	=	=	SYM
ap-7678	355	16	x6	x6	PROPN
ap-7678	355	17	−	−	PROPN
ap-7678	355	18	4x2	4x2	NUM
ap-7678	355	19	,	,	PUNCT
ap-7678	355	20	ms	ms	NOUN
ap-7678	355	21	=	=	SYM
ap-7678	355	22	5	5	NUM
ap-7678	355	23	,	,	PUNCT
ap-7678	355	24	rgauss	rgauss	NOUN
ap-7678	355	25	=	=	PUNCT
ap-7678	355	26	e−x2	e−x2	PUNCT
ap-7678	355	27	and	and	CCONJ
ap-7678	355	28	ra	ra	PROPN
ap-7678	355	29	=	=	PROPN
ap-7678	355	30	e−x4/4	e−x4/4	PROPN
ap-7678	355	31	.	.	PUNCT
ap-7678	356	1	n	n	PROPN
ap-7678	356	2	e5	e5	PROPN
ap-7678	356	3	e6	e6	PROPN
ap-7678	356	4	e7	e7	PROPN
ap-7678	356	5	e8	e8	PROPN
ap-7678	356	6	e9	e9	PROPN
ap-7678	356	7	25	25	NUM
ap-7678	356	8	23.5770262257055	23.5770262257055	NUM
ap-7678	356	9	46.5131514442214	46.5131514442214	NUM
ap-7678	356	10	50	50	NUM
ap-7678	356	11	23.8599097864908	23.8599097864908	NUM
ap-7678	356	12	31.6506317849753	31.6506317849753	NUM
ap-7678	356	13	40.1684238879451	40.1684238879451	NUM
ap-7678	356	14	48.7083019153321	48.7083019153321	NUM
ap-7678	356	15	57.2184300245252	57.2184300245252	NUM
ap-7678	356	16	75	75	NUM
ap-7678	356	17	23.8838452220218	23.8838452220218	NUM
ap-7678	356	18	31.7416917119713	31.7416917119713	NUM
ap-7678	356	19	40.2993336329482	40.2993336329482	NUM
ap-7678	356	20	49.5062748750452	49.5062748750452	NUM
ap-7678	356	21	59.2851256940478	59.2851256940478	NUM
ap-7678	356	22	100	100	NUM
ap-7678	356	23	23.8839209143498	23.8839209143498	NUM
ap-7678	356	24	31.7425529172254	31.7425529172254	NUM
ap-7678	356	25	40.3026888622591	40.3026888622591	NUM
ap-7678	356	26	49.5114653542220	49.5114653542220	NUM
ap-7678	356	27	59.3258556535193	59.3258556535193	NUM
ap-7678	356	28	125	125	NUM
ap-7678	356	29	23.8839223617171	23.8839223617171	NUM
ap-7678	356	30	31.7425500612024	31.7425500612024	NUM
ap-7678	356	31	40.3027376770457	40.3027376770457	NUM
ap-7678	356	32	49.5115009782085	49.5115009782085	NUM
ap-7678	356	33	59.3262796515870	59.3262796515870	NUM
ap-7678	356	34	150	150	NUM
ap-7678	356	35	23.8839223760489	23.8839223760489	NUM
ap-7678	356	36	31.7425498440774	31.7425498440774	NUM
ap-7678	356	37	40.3027378004240	40.3027378004240	NUM
ap-7678	356	38	49.5115004143879	49.5115004143879	NUM
ap-7678	357	1	59.3262744628777	59.3262744628777	NUM
ap-7678	357	2	175	175	NUM
ap-7678	357	3	23.8839223758291	23.8839223758291	NUM
ap-7678	357	4	31.7425498374411	31.7425498374411	NUM
ap-7678	357	5	40.3027377924283	40.3027377924283	NUM
ap-7678	357	6	49.5115003807374	49.5115003807374	NUM
ap-7678	357	7	59.3262741019580	59.3262741019580	NUM
ap-7678	357	8	200	200	NUM
ap-7678	357	9	23.8839223758112	23.8839223758112	NUM
ap-7678	357	10	31.7425498371238	31.7425498371238	NUM
ap-7678	357	11	40.3027377918922	40.3027377918922	NUM
ap-7678	357	12	49.5115003775253	49.5115003775253	NUM
ap-7678	357	13	59.3262740863702	59.3262740863702	NUM
ap-7678	357	14	25	25	NUM
ap-7678	357	15	23.8828743775059	23.8828743775059	NUM
ap-7678	357	16	31.7384980035506	31.7384980035506	NUM
ap-7678	357	17	40.3209134822543	40.3209134822543	NUM
ap-7678	357	18	50.0634788389675	50.0634788389675	NUM
ap-7678	357	19	50	50	NUM
ap-7678	357	20	23.8839223758082	23.8839223758082	NUM
ap-7678	357	21	31.7425498371192	31.7425498371192	NUM
ap-7678	357	22	40.3027377921622	40.3027377921622	NUM
ap-7678	357	23	49.5115003777545	49.5115003777545	NUM
ap-7678	357	24	59.3262740709658	59.3262740709658	NUM
ap-7678	357	25	100	100	NUM
ap-7678	357	26	23.8839223758101	23.8839223758101	NUM
ap-7678	357	27	31.7425498371122	31.7425498371122	NUM
ap-7678	357	28	40.3027377918721	40.3027377918721	NUM
ap-7678	357	29	49.5115003773799	49.5115003773799	NUM
ap-7678	357	30	59.3262740857373	59.3262740857373	NUM
ap-7678	357	31	150	150	NUM
ap-7678	357	32	23.8839223758101	23.8839223758101	NUM
ap-7678	357	33	31.7425498371122	31.7425498371122	NUM
ap-7678	357	34	40.3027377918721	40.3027377918721	NUM
ap-7678	357	35	49.5115003773799	49.5115003773799	NUM
ap-7678	357	36	59.3262740857373	59.3262740857373	NUM
ap-7678	357	37	200	200	NUM
ap-7678	357	38	23.8839223758101	23.8839223758101	NUM
ap-7678	357	39	31.7425498371122	31.7425498371122	NUM
ap-7678	357	40	40.3027377918721	40.3027377918721	NUM
ap-7678	357	41	49.5115003773799	49.5115003773799	NUM
ap-7678	357	42	59.3262740857373	59.3262740857373	NUM
ap-7678	357	43	table	table	NOUN
ap-7678	357	44	2	2	NUM
ap-7678	357	45	.	.	PUNCT
ap-7678	357	46	oppq	oppq	PROPN
ap-7678	357	47	-	-	PUNCT
ap-7678	357	48	am	am	PROPN
ap-7678	357	49	,	,	PUNCT
ap-7678	357	50	for	for	ADP
ap-7678	357	51	the	the	DET
ap-7678	357	52	sixth	sixth	ADJ
ap-7678	357	53	-	-	PUNCT
ap-7678	357	54	tenth	tenth	ADJ
ap-7678	357	55	states	state	NOUN
ap-7678	357	56	of	of	ADP
ap-7678	357	57	v	v	NOUN
ap-7678	357	58	(	(	PUNCT
ap-7678	357	59	x	x	NOUN
ap-7678	357	60	)	)	PUNCT
ap-7678	357	61	=	=	SYM
ap-7678	357	62	x6	x6	PROPN
ap-7678	357	63	−	−	PROPN
ap-7678	357	64	4x2	4x2	NUM
ap-7678	357	65	,	,	PUNCT
ap-7678	357	66	ms	ms	NOUN
ap-7678	357	67	=	=	SYM
ap-7678	357	68	5	5	NUM
ap-7678	357	69	,	,	PUNCT
ap-7678	357	70	rgauss	rgauss	NOUN
ap-7678	357	71	=	=	PUNCT
ap-7678	357	72	e−x2	e−x2	PUNCT
ap-7678	357	73	and	and	CCONJ
ap-7678	357	74	ra	ra	PROPN
ap-7678	357	75	=	=	PROPN
ap-7678	357	76	e−x4/4	e−x4/4	PROPN
ap-7678	357	77	.	.	PUNCT
ap-7678	358	1	such	such	ADJ
ap-7678	358	2	states	state	NOUN
ap-7678	358	3	,	,	PUNCT
ap-7678	358	4	yielding	yield	VERB
ap-7678	358	5	a	a	DET
ap-7678	358	6	reduced	reduce	VERB
ap-7678	358	7	finite	finite	ADJ
ap-7678	358	8	order	order	NOUN
ap-7678	358	9	difference	difference	NOUN
ap-7678	358	10	equation	equation	NOUN
ap-7678	358	11	for	for	ADP
ap-7678	358	12	the	the	DET
ap-7678	358	13	power	power	NOUN
ap-7678	358	14	moments	moment	NOUN
ap-7678	358	15	(	(	PUNCT
ap-7678	358	16	i.e.	i.e.	X
ap-7678	358	17	ms	ms	X
ap-7678	358	18	=	=	PROPN
ap-7678	358	19	2	2	NUM
ap-7678	358	20	)	)	PUNCT
ap-7678	358	21	.	.	PUNCT
ap-7678	359	1	the	the	DET
ap-7678	359	2	results	result	NOUN
ap-7678	359	3	produced	produce	VERB
ap-7678	359	4	bounds	bound	NOUN
ap-7678	359	5	to	to	ADP
ap-7678	359	6	the	the	DET
ap-7678	359	7	8th	8th	ADJ
ap-7678	359	8	decimal	decimal	ADJ
ap-7678	359	9	place	place	NOUN
ap-7678	359	10	for	for	ADP
ap-7678	359	11	the	the	DET
ap-7678	359	12	ground	ground	NOUN
ap-7678	359	13	state	state	NOUN
ap-7678	359	14	energy	energy	NOUN
ap-7678	359	15	.	.	PUNCT
ap-7678	360	1	we	we	PRON
ap-7678	360	2	obtain	obtain	VERB
ap-7678	360	3	the	the	DET
ap-7678	360	4	emm	emm	PROPN
ap-7678	360	5	bounds	bound	NOUN
ap-7678	360	6	:	:	PUNCT
ap-7678	360	7	−0.523268623844284	−0.523268623844284	NOUN
ap-7678	360	8	<	<	X
ap-7678	360	9	egr	egr	PROPN
ap-7678	360	10	<	<	X
ap-7678	360	11	−0.523268619253327	−0.523268619253327	PROPN
ap-7678	360	12	,	,	PUNCT
ap-7678	360	13	based	base	VERB
ap-7678	360	14	on	on	ADP
ap-7678	360	15	an	an	DET
ap-7678	360	16	expansion	expansion	NOUN
ap-7678	360	17	order	order	NOUN
ap-7678	360	18	of	of	ADP
ap-7678	360	19	approximately	approximately	ADV
ap-7678	360	20	29	29	NUM
ap-7678	360	21	power	power	NOUN
ap-7678	360	22	moments	moment	NOUN
ap-7678	360	23	(	(	PUNCT
ap-7678	360	24	i.e.	i.e.	X
ap-7678	360	25	{	{	PUNCT
ap-7678	360	26	µ(p)|0	µ(p)|0	PROPN
ap-7678	360	27	≤	≤	NOUN
ap-7678	360	28	p	p	NOUN
ap-7678	360	29	≤	≤	NUM
ap-7678	360	30	28	28	NUM
ap-7678	360	31	}	}	PUNCT
ap-7678	360	32	)	)	PUNCT
ap-7678	360	33	.	.	PUNCT
ap-7678	361	1	this	this	PRON
ap-7678	361	2	was	be	AUX
ap-7678	361	3	done	do	VERB
ap-7678	361	4	on	on	ADP
ap-7678	361	5	a	a	DET
ap-7678	361	6	simple	simple	ADJ
ap-7678	361	7	pc	pc	NOUN
ap-7678	361	8	with	with	ADP
ap-7678	361	9	about	about	ADV
ap-7678	361	10	14	14	NUM
ap-7678	361	11	place	place	NOUN
ap-7678	361	12	precision	precision	NOUN
ap-7678	361	13	.	.	PUNCT
ap-7678	362	1	we	we	PRON
ap-7678	362	2	note	note	VERB
ap-7678	362	3	that	that	SCONJ
ap-7678	362	4	the	the	DET
ap-7678	362	5	emm	emm	PROPN
ap-7678	362	6	bounds	bound	NOUN
ap-7678	362	7	quoted	quote	VERB
ap-7678	362	8	above	above	ADV
ap-7678	362	9	were	be	AUX
ap-7678	362	10	accurate	accurate	ADJ
ap-7678	362	11	to	to	ADP
ap-7678	362	12	approximately	approximately	ADV
ap-7678	362	13	eight	eight	NUM
ap-7678	362	14	decimal	decimal	ADJ
ap-7678	362	15	places	place	NOUN
ap-7678	362	16	,	,	PUNCT
ap-7678	362	17	based	base	VERB
ap-7678	362	18	on	on	ADP
ap-7678	362	19	approximately	approximately	ADV
ap-7678	362	20	28	28	NUM
ap-7678	362	21	-	-	SYM
ap-7678	362	22	29	29	NUM
ap-7678	362	23	power	power	NOUN
ap-7678	362	24	moments	moment	NOUN
ap-7678	362	25	.	.	PUNCT
ap-7678	363	1	the	the	DET
ap-7678	363	2	missing	missing	ADJ
ap-7678	363	3	moment	moment	NOUN
ap-7678	363	4	order	order	NOUN
ap-7678	363	5	used	use	VERB
ap-7678	363	6	for	for	ADP
ap-7678	363	7	the	the	DET
ap-7678	363	8	even	even	ADJ
ap-7678	363	9	state	state	NOUN
ap-7678	363	10	formulation	formulation	NOUN
ap-7678	363	11	was	be	AUX
ap-7678	363	12	ms	ms	NOUN
ap-7678	363	13	=	=	PROPN
ap-7678	363	14	2	2	NUM
ap-7678	363	15	.	.	PUNCT
ap-7678	364	1	the	the	DET
ap-7678	364	2	oppq	oppq	PROPN
ap-7678	364	3	-	-	PUNCT
ap-7678	364	4	bm	bm	PROPN
ap-7678	364	5	bounds	bound	NOUN
ap-7678	364	6	quoted	quote	VERB
ap-7678	364	7	in	in	ADP
ap-7678	364	8	table	table	NOUN
ap-7678	364	9	3	3	NUM
ap-7678	364	10	,	,	PUNCT
ap-7678	364	11	based	base	VERB
ap-7678	364	12	on	on	ADP
ap-7678	364	13	an	an	DET
ap-7678	364	14	ms	ms	NOUN
ap-7678	364	15	=	=	SYM
ap-7678	364	16	5	5	NUM
ap-7678	364	17	mer	mer	NOUN
ap-7678	364	18	formulation	formulation	NOUN
ap-7678	364	19	,	,	PUNCT
ap-7678	364	20	use	use	VERB
ap-7678	364	21	an	an	DET
ap-7678	364	22	optimal	optimal	ADJ
ap-7678	364	23	weight	weight	NOUN
ap-7678	364	24	,	,	PUNCT
ap-7678	364	25	but	but	CCONJ
ap-7678	364	26	only	only	ADV
ap-7678	364	27	give	give	VERB
ap-7678	364	28	us	we	PRON
ap-7678	364	29	two	two	NUM
ap-7678	364	30	decimal	decimal	ADJ
ap-7678	364	31	place	place	NOUN
ap-7678	364	32	accuracy	accuracy	NOUN
ap-7678	364	33	on	on	ADP
ap-7678	364	34	the	the	DET
ap-7678	364	35	basis	basis	NOUN
ap-7678	364	36	of	of	ADP
ap-7678	364	37	25	25	NUM
ap-7678	364	38	power	power	NOUN
ap-7678	364	39	moments	moment	NOUN
ap-7678	364	40	.	.	PUNCT
ap-7678	365	1	of	of	ADP
ap-7678	365	2	course	course	NOUN
ap-7678	365	3	,	,	PUNCT
ap-7678	365	4	the	the	DET
ap-7678	365	5	bounds	bound	NOUN
ap-7678	365	6	quickly	quickly	ADV
ap-7678	365	7	improve	improve	VERB
ap-7678	365	8	,	,	PUNCT
ap-7678	365	9	as	as	ADP
ap-7678	365	10	n	n	PROPN
ap-7678	365	11	→	→	SYM
ap-7678	365	12	125	125	NUM
ap-7678	365	13	.	.	PUNCT
ap-7678	366	1	we	we	PRON
ap-7678	366	2	continue	continue	VERB
ap-7678	366	3	these	these	DET
ap-7678	366	4	emm	emm	PROPN
ap-7678	366	5	versus	versus	ADP
ap-7678	366	6	oppq	oppq	PROPN
ap-7678	366	7	comparisons	comparison	NOUN
ap-7678	366	8	below	below	ADV
ap-7678	366	9	.	.	PUNCT
ap-7678	367	1	by	by	ADP
ap-7678	367	2	way	way	NOUN
ap-7678	367	3	of	of	ADP
ap-7678	367	4	contrast	contrast	NOUN
ap-7678	367	5	,	,	PUNCT
ap-7678	367	6	the	the	DET
ap-7678	367	7	oppq	oppq	PROPN
ap-7678	367	8	-	-	PUNCT
ap-7678	367	9	bm	bm	PROPN
ap-7678	367	10	procedure	procedure	NOUN
ap-7678	367	11	can	can	AUX
ap-7678	367	12	produce	produce	VERB
ap-7678	367	13	bounds	bound	NOUN
ap-7678	367	14	on	on	ADP
ap-7678	367	15	all	all	DET
ap-7678	367	16	the	the	DET
ap-7678	367	17	low	low	ADV
ap-7678	367	18	-	-	PUNCT
ap-7678	367	19	lying	lie	VERB
ap-7678	367	20	states	state	NOUN
ap-7678	367	21	.	.	PUNCT
ap-7678	368	1	to	to	PART
ap-7678	368	2	produce	produce	VERB
ap-7678	368	3	these	these	PRON
ap-7678	368	4	,	,	PUNCT
ap-7678	368	5	we	we	PRON
ap-7678	368	6	must	must	AUX
ap-7678	368	7	first	first	ADV
ap-7678	368	8	generate	generate	VERB
ap-7678	368	9	the	the	DET
ap-7678	368	10	local	local	ADJ
ap-7678	368	11	minima	minima	NOUN
ap-7678	368	12	,	,	PUNCT
ap-7678	368	13	∂eλn	∂eλn	PUNCT
ap-7678	368	14	(	(	PUNCT
ap-7678	368	15	e(min	e(min	NOUN
ap-7678	368	16	)	)	PUNCT
ap-7678	368	17	phys;n	phys;n	PROPN
ap-7678	368	18	)	)	PUNCT
ap-7678	369	1	=	=	SYM
ap-7678	369	2	0	0	X
ap-7678	369	3	.	.	PUNCT
ap-7678	370	1	fortunately	fortunately	ADV
ap-7678	370	2	,	,	PUNCT
ap-7678	370	3	these	these	DET
ap-7678	370	4	derivatives	derivative	NOUN
ap-7678	370	5	can	can	AUX
ap-7678	370	6	be	be	AUX
ap-7678	370	7	obtained	obtain	VERB
ap-7678	370	8	algebraically	algebraically	ADV
ap-7678	370	9	through	through	ADP
ap-7678	370	10	a	a	DET
ap-7678	370	11	recursion	recursion	NOUN
ap-7678	370	12	procedure	procedure	NOUN
ap-7678	370	13	.	.	PUNCT
ap-7678	371	1	following	follow	VERB
ap-7678	371	2	this	this	PRON
ap-7678	371	3	,	,	PUNCT
ap-7678	371	4	we	we	PRON
ap-7678	371	5	must	must	AUX
ap-7678	371	6	discern	discern	VERB
ap-7678	371	7	a	a	DET
ap-7678	371	8	crude	crude	ADJ
ap-7678	371	9	upper	upper	ADJ
ap-7678	371	10	bound	bind	VERB
ap-7678	371	11	(	(	PUNCT
ap-7678	371	12	i.e.	i.e.	X
ap-7678	371	13	b(u	b(u	PROPN
ap-7678	371	14	)	)	PUNCT
ap-7678	371	15	phys	phy	NOUN
ap-7678	371	16	)	)	PUNCT
ap-7678	371	17	to	to	ADP
ap-7678	371	18	the	the	DET
ap-7678	371	19	positive	positive	ADJ
ap-7678	371	20	sequence	sequence	NOUN
ap-7678	371	21	{	{	PUNCT
ap-7678	371	22	λn	λn	X
ap-7678	371	23	(	(	PUNCT
ap-7678	371	24	e(min	e(min	PROPN
ap-7678	371	25	)	)	PUNCT
ap-7678	371	26	phys;n	phys;n	PROPN
ap-7678	371	27	)	)	PUNCT
ap-7678	371	28	|n	|n	X
ap-7678	371	29	>	>	X
ap-7678	371	30	0	0	NUM
ap-7678	371	31	}	}	PUNCT
ap-7678	371	32	<	<	X
ap-7678	371	33	b(u	b(u	PROPN
ap-7678	371	34	)	)	PUNCT
ap-7678	371	35	phys	phy	NOUN
ap-7678	371	36	.	.	PUNCT
ap-7678	372	1	we	we	PRON
ap-7678	372	2	then	then	ADV
ap-7678	372	3	determine	determine	VERB
ap-7678	372	4	the	the	DET
ap-7678	372	5	energy	energy	NOUN
ap-7678	372	6	interval	interval	NOUN
ap-7678	372	7	whose	whose	DET
ap-7678	372	8	endpoints	endpoint	NOUN
ap-7678	372	9	satisfy	satisfy	VERB
ap-7678	372	10	λn	λn	PROPN
ap-7678	372	11	(	(	PUNCT
ap-7678	372	12	e(l	e(l	PROPN
ap-7678	372	13	)	)	PUNCT
ap-7678	372	14	phys;n	phys;n	PROPN
ap-7678	372	15	)	)	PUNCT
ap-7678	372	16	=	=	SYM
ap-7678	372	17	b(u	b(u	PROPN
ap-7678	372	18	)	)	PUNCT
ap-7678	372	19	phys	phy	NOUN
ap-7678	372	20	=	=	SYM
ap-7678	372	21	λn	λn	PROPN
ap-7678	372	22	(	(	PUNCT
ap-7678	372	23	e(u	e(u	PROPN
ap-7678	372	24	)	)	PUNCT
ap-7678	372	25	phys;n	phys;n	PROPN
ap-7678	372	26	)	)	PUNCT
ap-7678	372	27	.	.	PUNCT
ap-7678	373	1	these	these	PRON
ap-7678	373	2	become	become	VERB
ap-7678	373	3	the	the	DET
ap-7678	373	4	lower	low	ADJ
ap-7678	373	5	and	and	CCONJ
ap-7678	373	6	upper	upper	ADJ
ap-7678	373	7	bound	bind	VERB
ap-7678	373	8	estimates	estimate	NOUN
ap-7678	373	9	for	for	ADP
ap-7678	373	10	that	that	DET
ap-7678	373	11	physical	physical	ADJ
ap-7678	373	12	energy	energy	NOUN
ap-7678	373	13	.	.	PUNCT
ap-7678	374	1	the	the	DET
ap-7678	374	2	above	above	ADJ
ap-7678	374	3	bounding	bound	VERB
ap-7678	374	4	analysis	analysis	NOUN
ap-7678	374	5	is	be	AUX
ap-7678	374	6	initiated	initiate	VERB
ap-7678	374	7	in	in	ADP
ap-7678	374	8	table	table	NOUN
ap-7678	374	9	4	4	NUM
ap-7678	374	10	for	for	ADP
ap-7678	374	11	the	the	DET
ap-7678	374	12	ground	ground	NOUN
ap-7678	374	13	and	and	CCONJ
ap-7678	374	14	second	second	ADJ
ap-7678	374	15	excited	excited	ADJ
ap-7678	374	16	state	state	NOUN
ap-7678	374	17	of	of	ADP
ap-7678	374	18	the	the	DET
ap-7678	374	19	sextic	sextic	ADJ
ap-7678	374	20	anharmonic	anharmonic	ADJ
ap-7678	374	21	oscillator	oscillator	NOUN
ap-7678	374	22	.	.	PUNCT
ap-7678	375	1	since	since	SCONJ
ap-7678	375	2	the	the	DET
ap-7678	375	3	use	use	NOUN
ap-7678	375	4	of	of	ADP
ap-7678	375	5	the	the	DET
ap-7678	375	6	ra	ra	PROPN
ap-7678	375	7	weight	weight	NOUN
ap-7678	375	8	yields	yield	NOUN
ap-7678	375	9	very	very	ADV
ap-7678	375	10	rapid	rapid	ADJ
ap-7678	375	11	convergence	convergence	NOUN
ap-7678	375	12	,	,	PUNCT
ap-7678	375	13	we	we	PRON
ap-7678	375	14	see	see	VERB
ap-7678	375	15	that	that	SCONJ
ap-7678	375	16	the	the	DET
ap-7678	375	17	coarse	coarse	ADJ
ap-7678	375	18	upper	upper	ADJ
ap-7678	375	19	bounds	bound	NOUN
ap-7678	375	20	,	,	PUNCT
ap-7678	375	21	b(u	b(u	PROPN
ap-7678	375	22	)	)	PUNCT
ap-7678	375	23	are	be	AUX
ap-7678	375	24	easily	easily	ADV
ap-7678	375	25	determined	determine	VERB
ap-7678	375	26	.	.	PUNCT
ap-7678	376	1	using	use	VERB
ap-7678	376	2	these	these	PRON
ap-7678	376	3	we	we	PRON
ap-7678	376	4	can	can	AUX
ap-7678	376	5	generate	generate	VERB
ap-7678	376	6	the	the	DET
ap-7678	376	7	bounds	bound	NOUN
ap-7678	376	8	for	for	ADP
ap-7678	376	9	the	the	DET
ap-7678	376	10	ground	ground	NOUN
ap-7678	376	11	state	state	NOUN
ap-7678	376	12	and	and	CCONJ
ap-7678	376	13	first	first	ADJ
ap-7678	376	14	excited	excited	ADJ
ap-7678	376	15	state	state	NOUN
ap-7678	376	16	,	,	PUNCT
ap-7678	376	17	as	as	SCONJ
ap-7678	376	18	given	give	VERB
ap-7678	376	19	in	in	ADP
ap-7678	376	20	table	table	NOUN
ap-7678	376	21	3	3	NUM
ap-7678	376	22	.	.	PUNCT
ap-7678	377	1	we	we	PRON
ap-7678	377	2	note	note	VERB
ap-7678	377	3	that	that	SCONJ
ap-7678	377	4	the	the	DET
ap-7678	377	5	oppq	oppq	PROPN
ap-7678	377	6	-	-	PUNCT
ap-7678	377	7	bm	bm	PROPN
ap-7678	377	8	bounds	bound	NOUN
ap-7678	377	9	generated	generate	VERB
ap-7678	377	10	in	in	ADP
ap-7678	377	11	table	table	NOUN
ap-7678	377	12	3	3	NUM
ap-7678	377	13	(	(	PUNCT
ap-7678	377	14	we	we	PRON
ap-7678	377	15	could	could	AUX
ap-7678	377	16	have	have	AUX
ap-7678	377	17	continued	continue	VERB
ap-7678	377	18	tightening	tighten	VERB
ap-7678	377	19	the	the	DET
ap-7678	377	20	bounds	bound	NOUN
ap-7678	377	21	)	)	PUNCT
ap-7678	377	22	,	,	PUNCT
ap-7678	377	23	for	for	ADP
ap-7678	377	24	the	the	DET
ap-7678	377	25	ground	ground	NOUN
ap-7678	377	26	state	state	NOUN
ap-7678	377	27	,	,	PUNCT
ap-7678	377	28	used	use	VERB
ap-7678	377	29	125	125	NUM
ap-7678	377	30	power	power	NOUN
ap-7678	377	31	moments	moment	NOUN
ap-7678	377	32	giving	give	VERB
ap-7678	377	33	us	we	PRON
ap-7678	377	34	bounds	bound	NOUN
ap-7678	377	35	at	at	ADP
ap-7678	377	36	the	the	DET
ap-7678	377	37	18th	18th	ADJ
ap-7678	377	38	decimal	decimal	ADJ
ap-7678	377	39	place	place	NOUN
ap-7678	377	40	.	.	PUNCT
ap-7678	378	1	in	in	ADP
ap-7678	378	2	eq	eq	ADP
ap-7678	378	3	.	.	PUNCT
ap-7678	379	1	(	(	PUNCT
ap-7678	379	2	69	69	NUM
ap-7678	379	3	)	)	PUNCT
ap-7678	379	4	we	we	PRON
ap-7678	379	5	quote	quote	VERB
ap-7678	379	6	the	the	DET
ap-7678	379	7	emm	emm	PROPN
ap-7678	379	8	bounds	bound	NOUN
ap-7678	379	9	obtained	obtain	VERB
ap-7678	379	10	on	on	ADP
ap-7678	379	11	the	the	DET
ap-7678	379	12	basis	basis	NOUN
ap-7678	379	13	of	of	ADP
ap-7678	379	14	61	61	NUM
ap-7678	379	15	power	power	NOUN
ap-7678	379	16	moments	moment	NOUN
ap-7678	379	17	.	.	PUNCT
ap-7678	380	1	the	the	DET
ap-7678	380	2	accuracy	accuracy	NOUN
ap-7678	380	3	is	be	AUX
ap-7678	380	4	at	at	ADP
ap-7678	380	5	the	the	DET
ap-7678	380	6	33rd	33rd	ADJ
ap-7678	380	7	decimal	decimal	ADJ
ap-7678	380	8	place	place	NOUN
ap-7678	380	9	.	.	PUNCT
ap-7678	381	1	however	however	ADV
ap-7678	381	2	,	,	PUNCT
ap-7678	381	3	in	in	ADP
ap-7678	381	4	eq	eq	ADP
ap-7678	381	5	.	.	PUNCT
ap-7678	382	1	(	(	PUNCT
ap-7678	382	2	70	70	NUM
ap-7678	382	3	)	)	PUNCT
ap-7678	382	4	,	,	PUNCT
ap-7678	382	5	using	use	VERB
ap-7678	382	6	an	an	DET
ap-7678	382	7	optimal	optimal	ADJ
ap-7678	382	8	(	(	PUNCT
ap-7678	382	9	stieljes	stieljes	NOUN
ap-7678	382	10	representation	representation	NOUN
ap-7678	382	11	)	)	PUNCT
ap-7678	382	12	weight	weight	NOUN
ap-7678	382	13	,	,	PUNCT
ap-7678	382	14	the	the	DET
ap-7678	382	15	oppq	oppq	NOUN
ap-7678	382	16	-	-	PUNCT
ap-7678	382	17	am	am	NOUN
ap-7678	382	18	converged	converge	VERB
ap-7678	382	19	to	to	ADP
ap-7678	382	20	50	50	NUM
ap-7678	382	21	decimal	decimal	ADJ
ap-7678	382	22	places	place	NOUN
ap-7678	382	23	,	,	PUNCT
ap-7678	382	24	using	use	VERB
ap-7678	382	25	only	only	ADV
ap-7678	382	26	45	45	NUM
ap-7678	382	27	power	power	NOUN
ap-7678	382	28	moments	moment	NOUN
ap-7678	382	29	.	.	PUNCT
ap-7678	383	1	71	71	NUM
ap-7678	383	2	carlos	carlos	PROPN
ap-7678	383	3	r.	r.	PROPN
ap-7678	383	4	handy	handy	PROPN
ap-7678	383	5	acta	acta	PROPN
ap-7678	383	6	polytechnica	polytechnica	PROPN
ap-7678	383	7	n	n	PROPN
ap-7678	383	8	e	e	X
ap-7678	383	9	(	(	PUNCT
ap-7678	383	10	l	l	NOUN
ap-7678	383	11	)	)	PUNCT
ap-7678	383	12	0;n	0;n	NUM
ap-7678	383	13	e	e	X
ap-7678	383	14	(	(	PUNCT
ap-7678	383	15	u	u	NOUN
ap-7678	383	16	)	)	PUNCT
ap-7678	383	17	0;n	0;n	NUM
ap-7678	383	18	e	e	X
ap-7678	383	19	(	(	PUNCT
ap-7678	383	20	l	l	NOUN
ap-7678	383	21	)	)	PUNCT
ap-7678	383	22	2;n	2;n	NUM
ap-7678	383	23	e	e	NOUN
ap-7678	383	24	(	(	PUNCT
ap-7678	383	25	u	u	NOUN
ap-7678	383	26	)	)	PUNCT
ap-7678	383	27	2;n	2;n	NUM
ap-7678	383	28	25	25	NUM
ap-7678	383	29	-0.524852943468474	-0.524852943468474	NOUN
ap-7678	383	30	-0.521778245307431	-0.521778245307431	PUNCT
ap-7678	384	1	5.368449680815753	5.368449680815753	NUM
ap-7678	384	2	5.384374689469648	5.384374689469648	NUM
ap-7678	384	3	50	50	NUM
ap-7678	384	4	-0.523268749143883	-0.523268749143883	NOUN
ap-7678	384	5	-0.523268495112775	-0.523268495112775	NOUN
ap-7678	385	1	5.374969291406036	5.374969291406036	NUM
ap-7678	385	2	5.374970726288986	5.374970726288986	NUM
ap-7678	385	3	75	75	NUM
ap-7678	385	4	-0.523268622134517	-0.523268622134517	NOUN
ap-7678	385	5	-0.523268622120587	-0.523268622120587	NOUN
ap-7678	385	6	5.374970008804361	5.374970008804361	NUM
ap-7678	385	7	5.374970008875728	5.374970008875728	NUM
ap-7678	385	8	100	100	NUM
ap-7678	385	9	-0.523268622127553	-0.523268622127553	NOUN
ap-7678	385	10	-0.523268622127552	-0.523268622127552	NOUN
ap-7678	386	1	5.3749700088400432	5.3749700088400432	NUM
ap-7678	386	2	5.3749700088400468	5.3749700088400468	NUM
ap-7678	386	3	125	125	NUM
ap-7678	386	4	-0.52326862212755223948	-0.52326862212755223948	NOUN
ap-7678	386	5	-0.52326862212755223934	-0.52326862212755223934	NOUN
ap-7678	386	6	5.3749700088400449937	5.3749700088400449937	NUM
ap-7678	386	7	5.3749700088400449945	5.3749700088400449945	NUM
ap-7678	386	8	b(u	b(u	PROPN
ap-7678	386	9	)	)	PUNCT
ap-7678	386	10	=	=	PUNCT
ap-7678	386	11	−0.876	−0.876	ADP
ap-7678	386	12	b(u	b(u	PROPN
ap-7678	386	13	)	)	PUNCT
ap-7678	386	14	=	=	SYM
ap-7678	386	15	−0.990	−0.990	NOUN
ap-7678	386	16	table	table	NOUN
ap-7678	386	17	3	3	NUM
ap-7678	386	18	.	.	PUNCT
ap-7678	386	19	oppq	oppq	PROPN
ap-7678	386	20	-	-	PUNCT
ap-7678	386	21	bm	bm	X
ap-7678	386	22	upper	upper	ADJ
ap-7678	386	23	and	and	CCONJ
ap-7678	386	24	lower	low	ADJ
ap-7678	386	25	bounds	bound	NOUN
ap-7678	386	26	for	for	ADP
ap-7678	386	27	the	the	DET
ap-7678	386	28	ground	ground	NOUN
ap-7678	386	29	and	and	CCONJ
ap-7678	386	30	second	second	ADJ
ap-7678	386	31	excited	excited	ADJ
ap-7678	386	32	states	state	NOUN
ap-7678	386	33	,	,	PUNCT
ap-7678	386	34	using	use	VERB
ap-7678	386	35	ra	ra	PROPN
ap-7678	386	36	=	=	SYM
ap-7678	386	37	e−x4/4	e−x4/4	PROPN
ap-7678	386	38	.	.	PUNCT
ap-7678	386	39	n	n	PRON
ap-7678	386	40	∂eλ(e(min	∂eλ(e(min	NOUN
ap-7678	386	41	)	)	PUNCT
ap-7678	386	42	0;n	0;n	NOUN
ap-7678	386	43	)	)	PUNCT
ap-7678	386	44	=	=	SYM
ap-7678	386	45	0	0	NUM
ap-7678	386	46	log10	log10	PROPN
ap-7678	386	47	(	(	PUNCT
ap-7678	386	48	λ(e(min	λ(e(min	PROPN
ap-7678	386	49	)	)	PUNCT
ap-7678	386	50	0;n	0;n	NUM
ap-7678	386	51	)	)	PUNCT
ap-7678	386	52	)	)	PUNCT
ap-7678	387	1	∂eλ(e(min	∂eλ(e(min	ADJ
ap-7678	387	2	)	)	PUNCT
ap-7678	387	3	2;n	2;n	NUM
ap-7678	387	4	)	)	PUNCT
ap-7678	388	1	=	=	SYM
ap-7678	388	2	0	0	NUM
ap-7678	389	1	log10	log10	PROPN
ap-7678	389	2	(	(	PUNCT
ap-7678	389	3	λ(e(min	λ(e(min	PROPN
ap-7678	389	4	)	)	PUNCT
ap-7678	389	5	2;n	2;n	NUM
ap-7678	389	6	)	)	PUNCT
ap-7678	389	7	)	)	PUNCT
ap-7678	390	1	25	25	NUM
ap-7678	390	2	-0.523315367444853	-0.523315367444853	NOUN
ap-7678	390	3	-0.877054685910968	-0.877054685910968	X
ap-7678	390	4	5.37640170043752	5.37640170043752	NUM
ap-7678	390	5	-0.991603081677466	-0.991603081677466	PUNCT
ap-7678	390	6	50	50	NUM
ap-7678	390	7	-0.523268622128326	-0.523268622128326	NOUN
ap-7678	390	8	-0.877042389134943	-0.877042389134943	VERB
ap-7678	391	1	5.37497000884746	5.37497000884746	NUM
ap-7678	391	2	-0.991530473233468	-0.991530473233468	NOUN
ap-7678	391	3	75	75	NUM
ap-7678	391	4	-0.523268622127552	-0.523268622127552	NOUN
ap-7678	391	5	-0.877042389134815	-0.877042389134815	NOUN
ap-7678	391	6	5.37497000884004	5.37497000884004	NUM
ap-7678	391	7	-0.991530473233135	-0.991530473233135	NOUN
ap-7678	391	8	100	100	NUM
ap-7678	391	9	-0.523268622127552	-0.523268622127552	NOUN
ap-7678	391	10	-0.877042389134815	-0.877042389134815	NOUN
ap-7678	391	11	5.37497000884004	5.37497000884004	NUM
ap-7678	391	12	-0.991530473233135	-0.991530473233135	PUNCT
ap-7678	391	13	b(u	b(u	NOUN
ap-7678	391	14	)	)	PUNCT
ap-7678	391	15	=	=	PUNCT
ap-7678	391	16	−0.876	−0.876	ADP
ap-7678	391	17	b(u	b(u	PROPN
ap-7678	391	18	)	)	PUNCT
ap-7678	391	19	=	=	SYM
ap-7678	391	20	−0.990	−0.990	NOUN
ap-7678	391	21	table	table	NOUN
ap-7678	391	22	4	4	NUM
ap-7678	391	23	.	.	PUNCT
ap-7678	392	1	oppq	oppq	PROPN
ap-7678	392	2	-	-	PUNCT
ap-7678	392	3	bm	bm	PROPN
ap-7678	392	4	analysis	analysis	NOUN
ap-7678	392	5	for	for	ADP
ap-7678	392	6	determining	determine	VERB
ap-7678	392	7	coarse	coarse	ADJ
ap-7678	392	8	upper	upper	ADJ
ap-7678	392	9	bounds	bound	NOUN
ap-7678	392	10	,	,	PUNCT
ap-7678	392	11	b(u	b(u	PROPN
ap-7678	392	12	)	)	PUNCT
ap-7678	392	13	,	,	PUNCT
ap-7678	392	14	for	for	ADP
ap-7678	392	15	the	the	DET
ap-7678	392	16	ground	ground	NOUN
ap-7678	392	17	and	and	CCONJ
ap-7678	392	18	second	second	ADJ
ap-7678	392	19	excited	excited	ADJ
ap-7678	392	20	states	state	NOUN
ap-7678	392	21	,	,	PUNCT
ap-7678	392	22	using	use	VERB
ap-7678	392	23	ra	ra	PROPN
ap-7678	392	24	=	=	PUNCT
ap-7678	392	25	e−x4/4	e−x4/4	PROPN
ap-7678	392	26	.	.	PUNCT
ap-7678	393	1	9.2	9.2	NUM
ap-7678	393	2	.	.	PUNCT
ap-7678	394	1	emm	emm	PROPN
ap-7678	394	2	-	-	PUNCT
ap-7678	394	3	ψ2	ψ2	VERB
ap-7678	394	4	another	another	DET
ap-7678	394	5	mer	mer	NOUN
ap-7678	394	6	representation	representation	NOUN
ap-7678	394	7	for	for	ADP
ap-7678	394	8	the	the	DET
ap-7678	394	9	sextic	sextic	PROPN
ap-7678	394	10	anharmonic	anharmonic	ADJ
ap-7678	394	11	oscillator	oscillator	NOUN
ap-7678	394	12	is	be	AUX
ap-7678	394	13	possible	possible	ADJ
ap-7678	394	14	by	by	ADP
ap-7678	394	15	working	work	VERB
ap-7678	394	16	with	with	ADP
ap-7678	394	17	the	the	DET
ap-7678	394	18	probability	probability	NOUN
ap-7678	394	19	density	density	NOUN
ap-7678	394	20	,	,	PUNCT
ap-7678	394	21	s(x	s(x	PROPN
ap-7678	394	22	)	)	PUNCT
ap-7678	394	23	=	=	SYM
ap-7678	394	24	ψ2(x	ψ2(x	PROPN
ap-7678	394	25	)	)	PUNCT
ap-7678	394	26	.	.	PUNCT
ap-7678	395	1	it	it	PRON
ap-7678	395	2	is	be	AUX
ap-7678	395	3	easy	easy	ADJ
ap-7678	395	4	to	to	PART
ap-7678	395	5	show	show	VERB
ap-7678	395	6	that	that	SCONJ
ap-7678	395	7	the	the	DET
ap-7678	395	8	probability	probability	NOUN
ap-7678	395	9	density	density	NOUN
ap-7678	395	10	for	for	ADP
ap-7678	395	11	real	real	ADJ
ap-7678	395	12	potentials	potential	NOUN
ap-7678	395	13	satisfies	satisfy	VERB
ap-7678	395	14	a	a	DET
ap-7678	395	15	third	third	ADJ
ap-7678	395	16	order	order	NOUN
ap-7678	395	17	lode	lode	NOUN
ap-7678	395	18	.	.	PUNCT
ap-7678	396	1	for	for	ADP
ap-7678	396	2	the	the	DET
ap-7678	396	3	sextic	sextic	PROPN
ap-7678	396	4	anharmonic	anharmonic	ADJ
ap-7678	396	5	oscillator	oscillator	NOUN
ap-7678	396	6	problem	problem	NOUN
ap-7678	396	7	,	,	PUNCT
ap-7678	396	8	this	this	DET
ap-7678	396	9	3rd	3rd	ADJ
ap-7678	396	10	order	order	NOUN
ap-7678	396	11	lode	lode	NOUN
ap-7678	396	12	yields	yield	NOUN
ap-7678	396	13	a	a	DET
ap-7678	396	14	mer	mer	PROPN
ap-7678	396	15	relation	relation	NOUN
ap-7678	396	16	of	of	ADP
ap-7678	396	17	order	order	NOUN
ap-7678	396	18	3	3	NUM
ap-7678	396	19	(	(	PUNCT
ap-7678	396	20	i.e.	i.e.	X
ap-7678	396	21	ms	ms	X
ap-7678	396	22	=	=	PROPN
ap-7678	396	23	2	2	NUM
ap-7678	396	24	)	)	PUNCT
ap-7678	396	25	.	.	PUNCT
ap-7678	397	1	we	we	PRON
ap-7678	397	2	do	do	AUX
ap-7678	397	3	not	not	PART
ap-7678	397	4	give	give	VERB
ap-7678	397	5	the	the	DET
ap-7678	397	6	details	detail	NOUN
ap-7678	397	7	of	of	ADP
ap-7678	397	8	this	this	DET
ap-7678	397	9	analysis	analysis	NOUN
ap-7678	397	10	,	,	PUNCT
ap-7678	397	11	since	since	SCONJ
ap-7678	397	12	the	the	DET
ap-7678	397	13	following	follow	VERB
ap-7678	397	14	mer	mer	PROPN
ap-7678	397	15	representation	representation	NOUN
ap-7678	397	16	offers	offer	VERB
ap-7678	397	17	the	the	DET
ap-7678	397	18	easiest	easy	ADJ
ap-7678	397	19	oppq	oppq	ADJ
ap-7678	397	20	implementation	implementation	NOUN
ap-7678	397	21	.	.	PUNCT
ap-7678	398	1	the	the	DET
ap-7678	398	2	value	value	NOUN
ap-7678	398	3	of	of	ADP
ap-7678	398	4	emm	emm	PROPN
ap-7678	398	5	-	-	PUNCT
ap-7678	398	6	ψ2	ψ2	NOUN
ap-7678	398	7	is	be	AUX
ap-7678	398	8	that	that	SCONJ
ap-7678	398	9	we	we	PRON
ap-7678	398	10	can	can	AUX
ap-7678	398	11	bound	bind	VERB
ap-7678	398	12	the	the	DET
ap-7678	398	13	discrete	discrete	ADJ
ap-7678	398	14	states	state	NOUN
ap-7678	398	15	and	and	CCONJ
ap-7678	398	16	use	use	VERB
ap-7678	398	17	the	the	DET
ap-7678	398	18	results	result	NOUN
ap-7678	398	19	to	to	PART
ap-7678	398	20	gauge	gauge	VERB
ap-7678	398	21	the	the	DET
ap-7678	398	22	effectiveness	effectiveness	NOUN
ap-7678	398	23	of	of	ADP
ap-7678	398	24	oppq	oppq	NOUN
ap-7678	398	25	.	.	PUNCT
ap-7678	399	1	this	this	PRON
ap-7678	399	2	is	be	AUX
ap-7678	399	3	referenced	reference	VERB
ap-7678	399	4	below	below	ADV
ap-7678	399	5	.	.	PUNCT
ap-7678	400	1	9.3	9.3	NUM
ap-7678	400	2	.	.	PUNCT
ap-7678	400	3	emm	emm	PROPN
ap-7678	400	4	e−	e−	PROPN
ap-7678	401	1	x4	x4	PROPN
ap-7678	401	2	4	4	NUM
ap-7678	401	3	ψ(x	ψ(x	NOUN
ap-7678	401	4	)	)	PUNCT
ap-7678	401	5	the	the	DET
ap-7678	401	6	third	third	ADJ
ap-7678	401	7	mer	mer	PROPN
ap-7678	401	8	representation	representation	NOUN
ap-7678	401	9	is	be	AUX
ap-7678	401	10	obtained	obtain	VERB
ap-7678	401	11	through	through	ADP
ap-7678	401	12	the	the	DET
ap-7678	401	13	contact	contact	NOUN
ap-7678	401	14	transformation	transformation	NOUN
ap-7678	401	15	,	,	PUNCT
ap-7678	401	16	φ(x	φ(x	NOUN
ap-7678	401	17	)	)	PUNCT
ap-7678	401	18	=	=	SYM
ap-7678	401	19	exp(−x4	exp(−x4	NOUN
ap-7678	401	20	4	4	NUM
ap-7678	401	21	)	)	PUNCT
ap-7678	401	22	ψ(x	ψ(x	NOUN
ap-7678	401	23	)	)	PUNCT
ap-7678	401	24	.	.	PUNCT
ap-7678	402	1	(	(	PUNCT
ap-7678	402	2	63	63	NUM
ap-7678	402	3	)	)	PUNCT
ap-7678	402	4	since	since	SCONJ
ap-7678	402	5	this	this	PRON
ap-7678	402	6	involves	involve	VERB
ap-7678	402	7	a	a	DET
ap-7678	402	8	factor	factor	NOUN
ap-7678	402	9	identical	identical	ADJ
ap-7678	402	10	to	to	ADP
ap-7678	402	11	the	the	DET
ap-7678	402	12	dominant	dominant	ADJ
ap-7678	402	13	wkb	wkb	NOUN
ap-7678	402	14	asymptotic	asymptotic	ADJ
ap-7678	402	15	form	form	NOUN
ap-7678	402	16	for	for	ADP
ap-7678	402	17	the	the	DET
ap-7678	402	18	physical	physical	ADJ
ap-7678	402	19	states	state	NOUN
ap-7678	402	20	,	,	PUNCT
ap-7678	402	21	the	the	DET
ap-7678	402	22	mer	mer	PROPN
ap-7678	402	23	representation	representation	NOUN
ap-7678	402	24	for	for	ADP
ap-7678	402	25	φ	φ	PROPN
ap-7678	402	26	will	will	AUX
ap-7678	402	27	involve	involve	VERB
ap-7678	402	28	fewer	few	ADJ
ap-7678	402	29	missing	missing	ADJ
ap-7678	402	30	moments	moment	NOUN
ap-7678	402	31	(	(	PUNCT
ap-7678	402	32	i.e.	i.e.	X
ap-7678	402	33	none	none	NOUN
ap-7678	402	34	,	,	PUNCT
ap-7678	402	35	after	after	ADP
ap-7678	402	36	a	a	DET
ap-7678	402	37	normalization	normalization	NOUN
ap-7678	402	38	)	)	PUNCT
ap-7678	402	39	than	than	ADP
ap-7678	402	40	the	the	DET
ap-7678	402	41	mer	mer	PROPN
ap-7678	402	42	for	for	ADP
ap-7678	402	43	ψ	ψ	NOUN
ap-7678	402	44	,	,	PUNCT
ap-7678	402	45	as	as	SCONJ
ap-7678	402	46	given	give	VERB
ap-7678	402	47	in	in	ADP
ap-7678	402	48	eq	eq	ADP
ap-7678	402	49	.	.	PUNCT
ap-7678	403	1	(	(	PUNCT
ap-7678	403	2	48	48	NUM
ap-7678	403	3	)	)	PUNCT
ap-7678	403	4	,	,	PUNCT
ap-7678	403	5	involving	involve	VERB
ap-7678	403	6	five	five	NUM
ap-7678	403	7	(	(	PUNCT
ap-7678	403	8	5	5	NUM
ap-7678	403	9	)	)	PUNCT
ap-7678	403	10	missing	miss	VERB
ap-7678	403	11	moments	moment	NOUN
ap-7678	403	12	(	(	PUNCT
ap-7678	403	13	after	after	ADP
ap-7678	403	14	imposing	impose	VERB
ap-7678	403	15	a	a	DET
ap-7678	403	16	normalization	normalization	NOUN
ap-7678	403	17	)	)	PUNCT
ap-7678	403	18	.	.	PUNCT
ap-7678	404	1	this	this	PRON
ap-7678	404	2	is	be	AUX
ap-7678	404	3	desirable	desirable	ADJ
ap-7678	404	4	since	since	SCONJ
ap-7678	404	5	the	the	PRON
ap-7678	404	6	lower	low	ADJ
ap-7678	404	7	the	the	DET
ap-7678	404	8	missing	missing	ADJ
ap-7678	404	9	moment	moment	NOUN
ap-7678	404	10	order	order	NOUN
ap-7678	404	11	,	,	PUNCT
ap-7678	404	12	the	the	PRON
ap-7678	404	13	faster	fast	ADV
ap-7678	404	14	the	the	DET
ap-7678	404	15	convergence	convergence	NOUN
ap-7678	404	16	of	of	ADP
ap-7678	404	17	either	either	CCONJ
ap-7678	404	18	oppq	oppq	PROPN
ap-7678	404	19	or	or	CCONJ
ap-7678	404	20	emm	emm	PROPN
ap-7678	404	21	.	.	PUNCT
ap-7678	405	1	we	we	PRON
ap-7678	405	2	provide	provide	VERB
ap-7678	405	3	the	the	DET
ap-7678	405	4	details	detail	NOUN
ap-7678	405	5	of	of	ADP
ap-7678	405	6	both	both	DET
ap-7678	405	7	approaches	approach	NOUN
ap-7678	405	8	below	below	ADV
ap-7678	405	9	.	.	PUNCT
ap-7678	406	1	9.4	9.4	NUM
ap-7678	406	2	.	.	PUNCT
ap-7678	407	1	oppq	oppq	ADJ
ap-7678	407	2	analysis	analysis	NOUN
ap-7678	407	3	of	of	ADP
ap-7678	407	4	the	the	DET
ap-7678	407	5	(	(	PUNCT
ap-7678	407	6	ms	ms	NOUN
ap-7678	407	7	=	=	SYM
ap-7678	407	8	0	0	NUM
ap-7678	407	9	)	)	PUNCT
ap-7678	407	10	sextic	sextic	ADJ
ap-7678	407	11	anharmonic	anharmonic	ADJ
ap-7678	407	12	double	double	ADJ
ap-7678	407	13	well	well	NOUN
ap-7678	407	14	oscillator	oscillator	NOUN
ap-7678	407	15	the	the	DET
ap-7678	407	16	double	double	ADJ
ap-7678	407	17	well	well	ADV
ap-7678	407	18	anharmonic	anharmonic	ADJ
ap-7678	407	19	problem	problem	NOUN
ap-7678	407	20	of	of	ADP
ap-7678	407	21	interest	interest	NOUN
ap-7678	407	22	is	be	AUX
ap-7678	407	23	that	that	SCONJ
ap-7678	407	24	for	for	ADP
ap-7678	407	25	the	the	DET
ap-7678	407	26	potential	potential	ADJ
ap-7678	407	27	v	v	NOUN
ap-7678	407	28	(	(	PUNCT
ap-7678	407	29	x	x	NOUN
ap-7678	407	30	)	)	PUNCT
ap-7678	407	31	=	=	SYM
ap-7678	407	32	x6	x6	PROPN
ap-7678	407	33	−	−	PROPN
ap-7678	407	34	4x2	4x2	NUM
ap-7678	407	35	,	,	PUNCT
ap-7678	407	36	where	where	SCONJ
ap-7678	407	37	x	x	PUNCT
ap-7678	407	38	∈	∈	PROPN
ap-7678	407	39	ℜ.	ℜ.	VERB
ap-7678	407	40	the	the	DET
ap-7678	407	41	physical	physical	ADJ
ap-7678	407	42	solutions	solution	NOUN
ap-7678	407	43	must	must	AUX
ap-7678	407	44	die	die	VERB
ap-7678	407	45	off	off	ADP
ap-7678	407	46	,	,	PUNCT
ap-7678	407	47	asymptotically	asymptotically	ADV
ap-7678	407	48	,	,	PUNCT
ap-7678	407	49	according	accord	VERB
ap-7678	407	50	to	to	ADP
ap-7678	407	51	the	the	DET
ap-7678	407	52	dominant	dominant	ADJ
ap-7678	407	53	wkb	wkb	NOUN
ap-7678	407	54	expression	expression	NOUN
ap-7678	407	55	ψ(x	ψ(x	NOUN
ap-7678	407	56	)	)	PUNCT
ap-7678	407	57	∼	∼	NOUN
ap-7678	407	58	exp(−	exp(−	PROPN
ap-7678	407	59	x4	x4	PROPN
ap-7678	407	60	4	4	NUM
ap-7678	407	61	)	)	PUNCT
ap-7678	407	62	.	.	PUNCT
ap-7678	408	1	if	if	SCONJ
ap-7678	408	2	we	we	PRON
ap-7678	408	3	work	work	VERB
ap-7678	408	4	with	with	ADP
ap-7678	408	5	the	the	DET
ap-7678	408	6	contact	contact	NOUN
ap-7678	408	7	transformation	transformation	NOUN
ap-7678	408	8	in	in	ADP
ap-7678	408	9	eq	eq	ADP
ap-7678	408	10	.	.	PUNCT
ap-7678	409	1	(	(	PUNCT
ap-7678	409	2	63	63	NUM
ap-7678	409	3	)	)	PUNCT
ap-7678	409	4	φ(x	φ(x	NOUN
ap-7678	409	5	)	)	PUNCT
ap-7678	409	6	=	=	X
ap-7678	409	7	exp(−	exp(−	PROPN
ap-7678	409	8	x4	x4	PROPN
ap-7678	409	9	4	4	NUM
ap-7678	409	10	)	)	PUNCT
ap-7678	409	11	ψ(x	ψ(x	NOUN
ap-7678	409	12	)	)	PUNCT
ap-7678	409	13	,	,	PUNCT
ap-7678	409	14	we	we	PRON
ap-7678	409	15	note	note	VERB
ap-7678	409	16	that	that	SCONJ
ap-7678	409	17	the	the	DET
ap-7678	409	18	discrete	discrete	ADJ
ap-7678	409	19	states	state	NOUN
ap-7678	409	20	remain	remain	VERB
ap-7678	409	21	normalizable	normalizable	ADJ
ap-7678	409	22	and	and	CCONJ
ap-7678	409	23	exponentially	exponentially	ADV
ap-7678	409	24	bounded	bound	VERB
ap-7678	409	25	,	,	PUNCT
ap-7678	409	26	in	in	ADP
ap-7678	409	27	the	the	DET
ap-7678	409	28	φ	φ	PROPN
ap-7678	409	29	representation	representation	NOUN
ap-7678	409	30	.	.	PUNCT
ap-7678	410	1	unphysical	unphysical	ADJ
ap-7678	410	2	ψ	ψ	NOUN
ap-7678	410	3	configurations	configuration	NOUN
ap-7678	410	4	(	(	PUNCT
ap-7678	410	5	i.e.	i.e.	X
ap-7678	410	6	non	non	ADJ
ap-7678	410	7	-	-	ADJ
ap-7678	410	8	normalizable	normalizable	ADJ
ap-7678	410	9	due	due	ADP
ap-7678	410	10	to	to	ADP
ap-7678	410	11	their	their	PRON
ap-7678	410	12	exponentially	exponentially	ADV
ap-7678	410	13	unbounded	unbounded	ADJ
ap-7678	410	14	form	form	NOUN
ap-7678	410	15	in	in	ADP
ap-7678	410	16	one	one	NUM
ap-7678	410	17	or	or	CCONJ
ap-7678	410	18	both	both	CCONJ
ap-7678	410	19	asymptotic	asymptotic	ADJ
ap-7678	410	20	directions	direction	NOUN
ap-7678	410	21	)	)	PUNCT
ap-7678	410	22	map	map	NOUN
ap-7678	410	23	into	into	ADP
ap-7678	410	24	non	non	ADJ
ap-7678	410	25	-	-	ADJ
ap-7678	410	26	normalizable	normalizable	ADJ
ap-7678	410	27	φ	φ	PROPN
ap-7678	410	28	configurations	configuration	NOUN
ap-7678	410	29	.	.	PUNCT
ap-7678	411	1	the	the	DET
ap-7678	411	2	emm	emm	PROPN
ap-7678	411	3	formalism	formalism	NOUN
ap-7678	411	4	works	work	VERB
ap-7678	411	5	in	in	ADP
ap-7678	411	6	either	either	DET
ap-7678	411	7	representation	representation	NOUN
ap-7678	411	8	,	,	PUNCT
ap-7678	411	9	precisely	precisely	ADV
ap-7678	411	10	because	because	SCONJ
ap-7678	411	11	of	of	ADP
ap-7678	411	12	this	this	PRON
ap-7678	411	13	.	.	PUNCT
ap-7678	412	1	we	we	PRON
ap-7678	412	2	note	note	VERB
ap-7678	412	3	that	that	SCONJ
ap-7678	412	4	the	the	DET
ap-7678	412	5	power	power	NOUN
ap-7678	412	6	moments	moment	NOUN
ap-7678	412	7	for	for	ADP
ap-7678	412	8	exponentially	exponentially	ADV
ap-7678	412	9	bounded	bounded	ADJ
ap-7678	412	10	configurations	configuration	NOUN
ap-7678	412	11	exist	exist	VERB
ap-7678	412	12	;	;	PUNCT
ap-7678	412	13	whereas	whereas	SCONJ
ap-7678	412	14	they	they	PRON
ap-7678	412	15	become	become	VERB
ap-7678	412	16	infinite	infinite	ADJ
ap-7678	412	17	(	(	PUNCT
ap-7678	412	18	or	or	CCONJ
ap-7678	412	19	do	do	AUX
ap-7678	412	20	not	not	PART
ap-7678	412	21	exist	exist	VERB
ap-7678	412	22	)	)	PUNCT
ap-7678	412	23	for	for	ADP
ap-7678	412	24	unphysical	unphysical	ADJ
ap-7678	412	25	configurations	configuration	NOUN
ap-7678	412	26	.	.	PUNCT
ap-7678	413	1	the	the	DET
ap-7678	413	2	φ	φ	PROPN
ap-7678	413	3	configurations	configuration	NOUN
ap-7678	413	4	must	must	AUX
ap-7678	413	5	satisfy	satisfy	VERB
ap-7678	413	6	the	the	DET
ap-7678	413	7	differential	differential	ADJ
ap-7678	413	8	equation	equation	NOUN
ap-7678	413	9	φ′′(x	φ′′(x	NOUN
ap-7678	413	10	)	)	PUNCT
ap-7678	413	11	+	+	NUM
ap-7678	413	12	2x3φ′(x	2x3φ′(x	NUM
ap-7678	413	13	)	)	PUNCT
ap-7678	414	1	+	+	CCONJ
ap-7678	414	2	(	(	PUNCT
ap-7678	414	3	7x2	7x2	NUM
ap-7678	414	4	+	+	CCONJ
ap-7678	414	5	e)φ(x	e)φ(x	X
ap-7678	414	6	)	)	PUNCT
ap-7678	414	7	=	=	SYM
ap-7678	415	1	0	0	X
ap-7678	415	2	.	.	PUNCT
ap-7678	416	1	(	(	PUNCT
ap-7678	416	2	64	64	NUM
ap-7678	416	3	)	)	PUNCT
ap-7678	416	4	the	the	DET
ap-7678	416	5	hamburger	hamburger	NOUN
ap-7678	416	6	moment	moment	NOUN
ap-7678	416	7	(	(	PUNCT
ap-7678	416	8	i.e.	i.e.	X
ap-7678	416	9	µ(p	µ(p	ADJ
ap-7678	416	10	)	)	PUNCT
ap-7678	416	11	≡	≡	PROPN
ap-7678	416	12	∫	∫	PROPN
ap-7678	416	13	ℜ	ℜ	PROPN
ap-7678	416	14	dx	dx	PROPN
ap-7678	416	15	xpφ(x	xpφ(x	PROPN
ap-7678	416	16	)	)	PUNCT
ap-7678	416	17	)	)	PUNCT
ap-7678	416	18	equation	equation	NOUN
ap-7678	416	19	becomes	become	VERB
ap-7678	416	20	(	(	PUNCT
ap-7678	416	21	2p	2p	NUM
ap-7678	416	22	−	−	NOUN
ap-7678	416	23	1)µ(p	1)µ(p	NUM
ap-7678	416	24	+	+	CCONJ
ap-7678	416	25	2	2	X
ap-7678	416	26	)	)	PUNCT
ap-7678	416	27	=	=	NOUN
ap-7678	416	28	p(p	p(p	ADP
ap-7678	416	29	−	−	PROPN
ap-7678	416	30	1)µ(p	1)µ(p	NUM
ap-7678	416	31	−	−	NOUN
ap-7678	416	32	2	2	NUM
ap-7678	416	33	)	)	PUNCT
ap-7678	416	34	+	+	CCONJ
ap-7678	416	35	eµ(p	eµ(p	NUM
ap-7678	416	36	)	)	PUNCT
ap-7678	416	37	.	.	PUNCT
ap-7678	417	1	the	the	DET
ap-7678	417	2	even	even	ADJ
ap-7678	417	3	parity	parity	NOUN
ap-7678	417	4	states	state	NOUN
ap-7678	417	5	will	will	AUX
ap-7678	417	6	admit	admit	VERB
ap-7678	417	7	a	a	DET
ap-7678	417	8	mer	mer	NOUN
ap-7678	417	9	for	for	ADP
ap-7678	417	10	the	the	DET
ap-7678	417	11	even	even	ADJ
ap-7678	417	12	order	order	NOUN
ap-7678	417	13	power	power	NOUN
ap-7678	417	14	moments	moment	NOUN
ap-7678	417	15	,	,	PUNCT
ap-7678	417	16	µ(2ρ	µ(2ρ	NOUN
ap-7678	417	17	)	)	PUNCT
ap-7678	417	18	=	=	SYM
ap-7678	417	19	u(ρ	u(ρ	PROPN
ap-7678	417	20	)	)	PUNCT
ap-7678	417	21	or	or	CCONJ
ap-7678	417	22	(	(	PUNCT
ap-7678	417	23	4ρ	4ρ	NOUN
ap-7678	417	24	−	−	PROPN
ap-7678	418	1	1)u(ρ	1)u(ρ	NUM
ap-7678	419	1	+	+	CCONJ
ap-7678	419	2	1	1	X
ap-7678	419	3	)	)	PUNCT
ap-7678	419	4	=	=	SYM
ap-7678	420	1	2ρ(2ρ	2ρ(2ρ	NUM
ap-7678	420	2	−	−	NUM
ap-7678	420	3	1)u(ρ	1)u(ρ	NUM
ap-7678	420	4	−	−	NOUN
ap-7678	420	5	1	1	NUM
ap-7678	420	6	)	)	PUNCT
ap-7678	420	7	+	+	NUM
ap-7678	420	8	eu(ρ	eu(ρ	NUM
ap-7678	420	9	)	)	PUNCT
ap-7678	420	10	.	.	PUNCT
ap-7678	421	1	(	(	PUNCT
ap-7678	421	2	65	65	NUM
ap-7678	421	3	)	)	PUNCT
ap-7678	421	4	the	the	DET
ap-7678	421	5	odd	odd	ADJ
ap-7678	421	6	parity	parity	NOUN
ap-7678	421	7	states	state	NOUN
ap-7678	421	8	µ(2ρ	µ(2ρ	PUNCT
ap-7678	421	9	+	+	NUM
ap-7678	421	10	1	1	X
ap-7678	421	11	)	)	PUNCT
ap-7678	421	12	≡	≡	PROPN
ap-7678	421	13	ν(ρ	ν(ρ	X
ap-7678	421	14	)	)	PUNCT
ap-7678	421	15	will	will	AUX
ap-7678	421	16	satisfy	satisfy	VERB
ap-7678	421	17	the	the	DET
ap-7678	421	18	mer	mer	NOUN
ap-7678	421	19	:	:	PUNCT
ap-7678	421	20	(	(	PUNCT
ap-7678	421	21	4ρ	4ρ	NOUN
ap-7678	421	22	+	+	X
ap-7678	421	23	1)ν(ρ	1)ν(ρ	NUM
ap-7678	421	24	+	+	CCONJ
ap-7678	421	25	1	1	X
ap-7678	421	26	)	)	PUNCT
ap-7678	421	27	=	=	SYM
ap-7678	422	1	2ρ(2ρ	2ρ(2ρ	NUM
ap-7678	422	2	+	+	NUM
ap-7678	422	3	1)ν(ρ	1)ν(ρ	NUM
ap-7678	422	4	−	−	NOUN
ap-7678	422	5	1	1	NUM
ap-7678	422	6	)	)	PUNCT
ap-7678	422	7	+	+	NUM
ap-7678	422	8	eν(ρ	eν(ρ	NUM
ap-7678	422	9	)	)	PUNCT
ap-7678	422	10	.	.	PUNCT
ap-7678	423	1	(	(	PUNCT
ap-7678	423	2	66	66	NUM
ap-7678	423	3	)	)	PUNCT
ap-7678	423	4	we	we	PRON
ap-7678	423	5	note	note	VERB
ap-7678	423	6	that	that	SCONJ
ap-7678	423	7	the	the	DET
ap-7678	423	8	even	even	ADV
ap-7678	423	9	order	order	NOUN
ap-7678	423	10	hamburger	hamburger	NOUN
ap-7678	423	11	moments	moment	NOUN
ap-7678	423	12	satisfy	satisfy	VERB
ap-7678	423	13	:	:	PUNCT
ap-7678	423	14	µ(2ρ	µ(2ρ	X
ap-7678	423	15	)	)	PUNCT
ap-7678	423	16	=	=	SYM
ap-7678	423	17	u(ρ	u(ρ	PROPN
ap-7678	423	18	)	)	PUNCT
ap-7678	423	19	=	=	SYM
ap-7678	424	1	∫	∫	PROPN
ap-7678	425	1	+	+	NUM
ap-7678	425	2	∞	∞	PROPN
ap-7678	425	3	0	0	NUM
ap-7678	425	4	dξ	dξ	PROPN
ap-7678	425	5	ξρ−1/2φ	ξρ−1/2φ	PROPN
ap-7678	425	6	(	(	PUNCT
ap-7678	425	7	√	√	NUM
ap-7678	425	8	ξ	ξ	NUM
ap-7678	425	9	)	)	PUNCT
ap-7678	425	10	,	,	PUNCT
ap-7678	425	11	where	where	SCONJ
ap-7678	425	12	x2	x2	PROPN
ap-7678	425	13	=	=	SYM
ap-7678	425	14	ξ	ξ	PROPN
ap-7678	425	15	;	;	PUNCT
ap-7678	425	16	however	however	ADV
ap-7678	425	17	,	,	PUNCT
ap-7678	425	18	the	the	DET
ap-7678	425	19	odd	odd	ADJ
ap-7678	425	20	order	order	NOUN
ap-7678	425	21	hamburger	hamburger	NOUN
ap-7678	425	22	moments	moment	NOUN
ap-7678	425	23	satisfy	satisfy	VERB
ap-7678	425	24	µ(2ρ	µ(2ρ	PUNCT
ap-7678	425	25	+	+	NUM
ap-7678	425	26	1	1	X
ap-7678	425	27	)	)	PUNCT
ap-7678	425	28	=	=	SYM
ap-7678	425	29	ν(ρ	ν(ρ	X
ap-7678	425	30	)	)	PUNCT
ap-7678	425	31	=	=	SYM
ap-7678	426	1	∫	∫	PROPN
ap-7678	427	1	+	+	NUM
ap-7678	427	2	∞	∞	PROPN
ap-7678	427	3	0	0	NUM
ap-7678	427	4	dξ	dξ	PROPN
ap-7678	427	5	ξρφ	ξρφ	PROPN
ap-7678	427	6	(	(	PUNCT
ap-7678	427	7	√	√	NOUN
ap-7678	427	8	ξ	ξ	NUM
ap-7678	427	9	)	)	PUNCT
ap-7678	427	10	.	.	PUNCT
ap-7678	428	1	the	the	DET
ap-7678	428	2	importance	importance	NOUN
ap-7678	428	3	of	of	ADP
ap-7678	428	4	these	these	DET
ap-7678	428	5	relations	relation	NOUN
ap-7678	428	6	is	be	AUX
ap-7678	428	7	that	that	SCONJ
ap-7678	428	8	the	the	DET
ap-7678	428	9	respective	respective	ADJ
ap-7678	428	10	72	72	NUM
ap-7678	428	11	vol	vol	NOUN
ap-7678	428	12	.	.	PUNCT
ap-7678	429	1	62	62	NUM
ap-7678	429	2	no	no	INTJ
ap-7678	429	3	.	.	PUNCT
ap-7678	430	1	1/2022	1/2022	NUM
ap-7678	430	2	algebraic	algebraic	ADJ
ap-7678	430	3	eigenenergy	eigenenergy	NOUN
ap-7678	430	4	bounding	bounding	NOUN
ap-7678	430	5	method	method	NOUN
ap-7678	430	6	n	n	PRON
ap-7678	430	7	e0	e0	PROPN
ap-7678	430	8	e2	e2	PROPN
ap-7678	430	9	e4	e4	PROPN
ap-7678	430	10	e6	e6	PROPN
ap-7678	430	11	e8	e8	PROPN
ap-7678	430	12	10	10	NUM
ap-7678	430	13	-0.5234534028	-0.5234534028	NOUN
ap-7678	430	14	5.354166238	5.354166238	NUM
ap-7678	430	15	16.82887249	16.82887249	NUM
ap-7678	430	16	20	20	NUM
ap-7678	430	17	-0.5232677208	-0.5232677208	VERB
ap-7678	430	18	5.375031586	5.375031586	NUM
ap-7678	430	19	16.79298386	16.79298386	NUM
ap-7678	430	20	31.63171200	31.63171200	NUM
ap-7678	430	21	49.94322409	49.94322409	NUM
ap-7678	430	22	30	30	NUM
ap-7678	430	23	-0.5232686293	-0.5232686293	NOUN
ap-7678	431	1	5.374969341	5.374969341	NUM
ap-7678	431	2	16.79536744	16.79536744	NUM
ap-7678	431	3	31.74329539	31.74329539	NUM
ap-7678	431	4	49.45632020	49.45632020	NUM
ap-7678	431	5	40	40	NUM
ap-7678	431	6	-0.5232686220	-0.5232686220	PROPN
ap-7678	431	7	5.374970019	5.374970019	NUM
ap-7678	431	8	16.79534604	16.79534604	NUM
ap-7678	431	9	31.74254727	31.74254727	NUM
ap-7678	431	10	49.51247497	49.51247497	NUM
ap-7678	431	11	50	50	NUM
ap-7678	431	12	-0.5232686221	-0.5232686221	NOUN
ap-7678	431	13	5.374970009	5.374970009	NUM
ap-7678	431	14	16.79534686	16.79534686	NUM
ap-7678	431	15	31.74254902	31.74254902	NUM
ap-7678	431	16	49.51148774	49.51148774	NUM
ap-7678	431	17	60	60	NUM
ap-7678	432	1	-0.5232686221	-0.5232686221	NOUN
ap-7678	433	1	5.374970009	5.374970009	NUM
ap-7678	433	2	16.79534683	16.79534683	NUM
ap-7678	433	3	31.74254987	31.74254987	NUM
ap-7678	433	4	49.51149946	49.51149946	NUM
ap-7678	433	5	70	70	NUM
ap-7678	433	6	-0.5232686221	-0.5232686221	NOUN
ap-7678	433	7	5.374970009	5.374970009	NUM
ap-7678	433	8	16.79534683	16.79534683	NUM
ap-7678	433	9	31.74254984	31.74254984	NUM
ap-7678	433	10	49.51150043	49.51150043	NUM
ap-7678	433	11	80	80	NUM
ap-7678	433	12	-0.5232686221	-0.5232686221	NOUN
ap-7678	433	13	5.374970009	5.374970009	NUM
ap-7678	433	14	16.79534683	16.79534683	NUM
ap-7678	433	15	31.74254984	31.74254984	NUM
ap-7678	433	16	49.51150038	49.51150038	NUM
ap-7678	433	17	90	90	NUM
ap-7678	433	18	-0.5232686221	-0.5232686221	NOUN
ap-7678	433	19	5.374970009	5.374970009	NUM
ap-7678	433	20	16.79534683	16.79534683	NUM
ap-7678	433	21	31.74254984	31.74254984	NUM
ap-7678	433	22	49.51150038	49.51150038	NUM
ap-7678	433	23	100	100	NUM
ap-7678	433	24	-0.5232686221	-0.5232686221	NOUN
ap-7678	433	25	5.374970009	5.374970009	NUM
ap-7678	433	26	16.79534683	16.79534683	NUM
ap-7678	433	27	31.74254984	31.74254984	NUM
ap-7678	433	28	49.51150038	49.51150038	NUM
ap-7678	433	29	table	table	NOUN
ap-7678	433	30	5	5	NUM
ap-7678	433	31	.	.	PUNCT
ap-7678	433	32	oppq	oppq	PROPN
ap-7678	433	33	-	-	PUNCT
ap-7678	433	34	bm	bm	PROPN
ap-7678	433	35	estimates	estimate	NOUN
ap-7678	433	36	(	(	PUNCT
ap-7678	433	37	i.e.	i.e.	X
ap-7678	433	38	∂eλn	∂eλn	PUNCT
ap-7678	433	39	(	(	PUNCT
ap-7678	433	40	e(min	e(min	NOUN
ap-7678	433	41	)	)	PUNCT
ap-7678	433	42	phys;n	phys;n	PROPN
ap-7678	433	43	)	)	PUNCT
ap-7678	433	44	=	=	SYM
ap-7678	433	45	0	0	NUM
ap-7678	433	46	)	)	PUNCT
ap-7678	433	47	,	,	PUNCT
ap-7678	433	48	φ	φ	PROPN
ap-7678	433	49	representation	representation	NOUN
ap-7678	433	50	,	,	PUNCT
ap-7678	433	51	(	(	PUNCT
ap-7678	433	52	eq	eq	NOUN
ap-7678	433	53	.	.	PROPN
ap-7678	433	54	65	65	NUM
ap-7678	433	55	)	)	PUNCT
ap-7678	433	56	rl(ξ	rl(ξ	PUNCT
ap-7678	433	57	)	)	PUNCT
ap-7678	433	58	=	=	SYM
ap-7678	434	1	ξ−	ξ−	NOUN
ap-7678	434	2	1	1	NUM
ap-7678	434	3	2	2	NUM
ap-7678	434	4	exp(−ξ	exp(−ξ	PROPN
ap-7678	434	5	)	)	PUNCT
ap-7678	434	6	.	.	PUNCT
ap-7678	435	1	moments	moment	NOUN
ap-7678	435	2	,	,	PUNCT
ap-7678	435	3	in	in	ADP
ap-7678	435	4	the	the	DET
ap-7678	435	5	context	context	NOUN
ap-7678	435	6	of	of	ADP
ap-7678	435	7	eqs	eqs	PROPN
ap-7678	435	8	.	.	PUNCT
ap-7678	436	1	(	(	PUNCT
ap-7678	436	2	65	65	NUM
ap-7678	436	3	,	,	PUNCT
ap-7678	436	4	66	66	NUM
ap-7678	436	5	)	)	PUNCT
ap-7678	436	6	allow	allow	VERB
ap-7678	436	7	us	we	PRON
ap-7678	436	8	to	to	PART
ap-7678	436	9	implement	implement	VERB
ap-7678	436	10	a	a	DET
ap-7678	436	11	stieltjes	stieltjes	NOUN
ap-7678	436	12	moment	moment	NOUN
ap-7678	436	13	analysis	analysis	NOUN
ap-7678	436	14	through	through	ADP
ap-7678	436	15	emm	emm	PROPN
ap-7678	436	16	.	.	PUNCT
ap-7678	437	1	we	we	PRON
ap-7678	437	2	will	will	AUX
ap-7678	437	3	only	only	ADV
ap-7678	437	4	apply	apply	VERB
ap-7678	437	5	oppq	oppq	NOUN
ap-7678	437	6	on	on	ADP
ap-7678	437	7	the	the	DET
ap-7678	437	8	even	even	ADJ
ap-7678	437	9	ψ	ψ	NOUN
ap-7678	437	10	configurations	configuration	NOUN
ap-7678	437	11	,	,	PUNCT
ap-7678	437	12	for	for	ADP
ap-7678	437	13	simplicity	simplicity	NOUN
ap-7678	437	14	.	.	PUNCT
ap-7678	438	1	however	however	ADV
ap-7678	438	2	,	,	PUNCT
ap-7678	438	3	since	since	SCONJ
ap-7678	438	4	the	the	DET
ap-7678	438	5	even	even	ADJ
ap-7678	438	6	order	order	NOUN
ap-7678	438	7	power	power	NOUN
ap-7678	438	8	moments	moment	NOUN
ap-7678	438	9	are	be	AUX
ap-7678	438	10	the	the	DET
ap-7678	438	11	moments	moment	NOUN
ap-7678	438	12	of	of	ADP
ap-7678	438	13	υ(ξ	υ(ξ	PROPN
ap-7678	438	14	)	)	PUNCT
ap-7678	438	15	≡	≡	PROPN
ap-7678	438	16	φ	φ	PROPN
ap-7678	438	17	(	(	PUNCT
ap-7678	438	18	√	√	PROPN
ap-7678	438	19	ξ)√	ξ)√	NUM
ap-7678	438	20	ξ	ξ	PROPN
ap-7678	438	21	,	,	PUNCT
ap-7678	438	22	the	the	DET
ap-7678	438	23	oppq	oppq	ADJ
ap-7678	438	24	expansion	expansion	NOUN
ap-7678	438	25	must	must	AUX
ap-7678	438	26	be	be	AUX
ap-7678	438	27	relative	relative	ADJ
ap-7678	438	28	to	to	ADP
ap-7678	438	29	this	this	DET
ap-7678	438	30	configuration	configuration	NOUN
ap-7678	438	31	.	.	PUNCT
ap-7678	439	1	thus	thus	ADV
ap-7678	439	2	,	,	PUNCT
ap-7678	439	3	the	the	DET
ap-7678	439	4	relevant	relevant	ADJ
ap-7678	439	5	oppq	oppq	ADJ
ap-7678	439	6	expansion	expansion	NOUN
ap-7678	439	7	will	will	AUX
ap-7678	439	8	be	be	AUX
ap-7678	439	9	υ(ξ	υ(ξ	PROPN
ap-7678	439	10	)	)	PUNCT
ap-7678	440	1	=	=	PUNCT
ap-7678	441	1	∞∑	∞∑	PRON
ap-7678	441	2	n=0	n=0	NUM
ap-7678	441	3	cnpn(ξ)r(ξ	cnpn(ξ)r(ξ	NOUN
ap-7678	441	4	)	)	PUNCT
ap-7678	441	5	,	,	PUNCT
ap-7678	441	6	(	(	PUNCT
ap-7678	441	7	67	67	NUM
ap-7678	441	8	)	)	PUNCT
ap-7678	441	9	where	where	SCONJ
ap-7678	441	10	φ	φ	PROPN
ap-7678	441	11	’s	’s	PART
ap-7678	441	12	power	power	NOUN
ap-7678	441	13	moments	moment	NOUN
ap-7678	441	14	will	will	AUX
ap-7678	441	15	satisfy	satisfy	VERB
ap-7678	441	16	eq	eq	ADP
ap-7678	441	17	.	.	PUNCT
ap-7678	442	1	(	(	PUNCT
ap-7678	442	2	65	65	NUM
ap-7678	442	3	)	)	PUNCT
ap-7678	442	4	.	.	PUNCT
ap-7678	443	1	since	since	SCONJ
ap-7678	443	2	the	the	DET
ap-7678	443	3	physical	physical	ADJ
ap-7678	443	4	configurations	configuration	NOUN
ap-7678	443	5	behave	behave	VERB
ap-7678	443	6	,	,	PUNCT
ap-7678	443	7	asymptotically	asymptotically	ADV
ap-7678	443	8	,	,	PUNCT
ap-7678	443	9	as	as	ADP
ap-7678	443	10	ψ(x	ψ(x	NOUN
ap-7678	443	11	)	)	PUNCT
ap-7678	443	12	∼	∼	NOUN
ap-7678	443	13	exp(−	exp(−	PROPN
ap-7678	443	14	x4	x4	PROPN
ap-7678	443	15	4	4	NUM
ap-7678	443	16	)	)	PUNCT
ap-7678	443	17	,	,	PUNCT
ap-7678	443	18	the	the	DET
ap-7678	443	19	transformed	transform	VERB
ap-7678	443	20	expressions	expression	NOUN
ap-7678	443	21	behave	behave	VERB
ap-7678	443	22	as	as	ADP
ap-7678	443	23	φ(ξ	φ(ξ	NOUN
ap-7678	443	24	)	)	PUNCT
ap-7678	443	25	∼	∼	NOUN
ap-7678	443	26	exp(−	exp(−	ADJ
ap-7678	443	27	ξ2	ξ2	NOUN
ap-7678	443	28	2	2	NUM
ap-7678	443	29	)	)	PUNCT
ap-7678	443	30	.	.	PUNCT
ap-7678	444	1	since	since	SCONJ
ap-7678	444	2	the	the	DET
ap-7678	444	3	transformed	transform	VERB
ap-7678	444	4	system	system	NOUN
ap-7678	444	5	involves	involve	VERB
ap-7678	444	6	a	a	DET
ap-7678	444	7	stieltjes	stieltjes	NOUN
ap-7678	444	8	configuration	configuration	NOUN
ap-7678	444	9	supported	support	VERB
ap-7678	444	10	on	on	ADP
ap-7678	444	11	the	the	DET
ap-7678	444	12	nonnegative	nonnegative	ADJ
ap-7678	444	13	axis	axis	NOUN
ap-7678	444	14	,	,	PUNCT
ap-7678	444	15	one	one	PRON
ap-7678	444	16	might	might	AUX
ap-7678	444	17	take	take	VERB
ap-7678	444	18	the	the	DET
ap-7678	444	19	weight	weight	NOUN
ap-7678	444	20	to	to	PART
ap-7678	444	21	be	be	AUX
ap-7678	444	22	the	the	DET
ap-7678	444	23	exponential	exponential	ADJ
ap-7678	444	24	function	function	NOUN
ap-7678	444	25	,	,	PUNCT
ap-7678	444	26	r̃l(ξ	r̃l(ξ	NOUN
ap-7678	444	27	)	)	PUNCT
ap-7678	444	28	=	=	SYM
ap-7678	444	29	exp(−ξ	exp(−ξ	PROPN
ap-7678	444	30	)	)	PUNCT
ap-7678	444	31	,	,	PUNCT
ap-7678	444	32	with	with	ADP
ap-7678	444	33	laguerre	laguerre	NOUN
ap-7678	444	34	polynomials	polynomial	NOUN
ap-7678	444	35	;	;	PUNCT
ap-7678	444	36	or	or	CCONJ
ap-7678	444	37	the	the	DET
ap-7678	444	38	gaussian	gaussian	NOUN
ap-7678	444	39	r̃g(ξ	r̃g(ξ	NOUN
ap-7678	444	40	)	)	PUNCT
ap-7678	444	41	=	=	NOUN
ap-7678	444	42	exp(−ξ2/2	exp(−ξ2/2	PRON
ap-7678	444	43	)	)	PUNCT
ap-7678	444	44	,	,	PUNCT
ap-7678	444	45	restricted	restrict	VERB
ap-7678	444	46	to	to	ADP
ap-7678	444	47	the	the	DET
ap-7678	444	48	nonnegative	nonnegative	ADJ
ap-7678	444	49	real	real	ADJ
ap-7678	444	50	axis	axis	NOUN
ap-7678	444	51	.	.	PUNCT
ap-7678	445	1	however	however	ADV
ap-7678	445	2	,	,	PUNCT
ap-7678	445	3	for	for	SCONJ
ap-7678	445	4	the	the	DET
ap-7678	445	5	quantization	quantization	NOUN
ap-7678	445	6	integral	integral	ADJ
ap-7678	445	7	in	in	ADP
ap-7678	445	8	eq	eq	ADP
ap-7678	445	9	.	.	PUNCT
ap-7678	446	1	(	(	PUNCT
ap-7678	446	2	20	20	NUM
ap-7678	446	3	)	)	PUNCT
ap-7678	446	4	to	to	PART
ap-7678	446	5	apply	apply	VERB
ap-7678	446	6	,	,	PUNCT
ap-7678	446	7	particularly	particularly	ADV
ap-7678	446	8	with	with	ADP
ap-7678	446	9	regards	regard	NOUN
ap-7678	446	10	to	to	ADP
ap-7678	446	11	the	the	DET
ap-7678	446	12	generation	generation	NOUN
ap-7678	446	13	of	of	ADP
ap-7678	446	14	bounds	bound	NOUN
ap-7678	446	15	,	,	PUNCT
ap-7678	446	16	we	we	PRON
ap-7678	446	17	need	need	VERB
ap-7678	446	18	to	to	PART
ap-7678	446	19	take	take	VERB
ap-7678	446	20	into	into	ADP
ap-7678	446	21	account	account	NOUN
ap-7678	446	22	the	the	DET
ap-7678	446	23	ξ−	ξ−	PROPN
ap-7678	446	24	1	1	NUM
ap-7678	446	25	2	2	NUM
ap-7678	446	26	that	that	PRON
ap-7678	446	27	is	be	AUX
ap-7678	446	28	inherent	inherent	ADJ
ap-7678	446	29	to	to	ADP
ap-7678	446	30	the	the	DET
ap-7678	446	31	transformed	transform	VERB
ap-7678	446	32	,	,	PUNCT
ap-7678	446	33	even	even	ADV
ap-7678	446	34	order	order	NOUN
ap-7678	446	35	,	,	PUNCT
ap-7678	446	36	power	power	NOUN
ap-7678	446	37	moments	moment	NOUN
ap-7678	446	38	.	.	PUNCT
ap-7678	447	1	based	base	VERB
ap-7678	447	2	on	on	ADP
ap-7678	447	3	the	the	DET
ap-7678	447	4	previous	previous	ADJ
ap-7678	447	5	arguments	argument	NOUN
ap-7678	447	6	we	we	PRON
ap-7678	447	7	will	will	AUX
ap-7678	447	8	work	work	VERB
ap-7678	447	9	with	with	ADP
ap-7678	447	10	rl(ξ	rl(ξ	PUNCT
ap-7678	447	11	)	)	PUNCT
ap-7678	447	12	=	=	SYM
ap-7678	447	13	ξ−	ξ−	NOUN
ap-7678	447	14	1	1	NUM
ap-7678	447	15	2	2	NUM
ap-7678	447	16	exp(−ξ	exp(−ξ	PROPN
ap-7678	447	17	)	)	PUNCT
ap-7678	447	18	and	and	CCONJ
ap-7678	447	19	rg(ξ	rg(ξ	PUNCT
ap-7678	447	20	)	)	PUNCT
ap-7678	448	1	=	=	SYM
ap-7678	448	2	ξ−	ξ−	NOUN
ap-7678	448	3	1	1	NUM
ap-7678	448	4	2	2	NUM
ap-7678	448	5	exp(−ξ2/2	exp(−ξ2/2	PRON
ap-7678	448	6	)	)	PUNCT
ap-7678	448	7	.	.	PUNCT
ap-7678	449	1	the	the	DET
ap-7678	449	2	corresponding	correspond	VERB
ap-7678	449	3	power	power	NOUN
ap-7678	449	4	moments	moment	NOUN
ap-7678	449	5	for	for	ADP
ap-7678	449	6	the	the	DET
ap-7678	449	7	weights	weight	NOUN
ap-7678	449	8	,	,	PUNCT
ap-7678	449	9	in	in	ADP
ap-7678	449	10	order	order	NOUN
ap-7678	449	11	to	to	PART
ap-7678	449	12	generate	generate	VERB
ap-7678	449	13	the	the	DET
ap-7678	449	14	orthonormal	orthonormal	ADJ
ap-7678	449	15	polynomials	polynomial	NOUN
ap-7678	449	16	are	be	AUX
ap-7678	449	17	obtained	obtain	VERB
ap-7678	449	18	as	as	ADP
ap-7678	449	19	follows	follow	VERB
ap-7678	449	20	.	.	PUNCT
ap-7678	450	1	the	the	DET
ap-7678	450	2	power	power	NOUN
ap-7678	450	3	moments	moment	NOUN
ap-7678	450	4	wl(p	wl(p	PUNCT
ap-7678	450	5	)	)	PUNCT
ap-7678	450	6	≡	≡	PROPN
ap-7678	450	7	∫	∫	PROPN
ap-7678	451	1	∞	∞	PROPN
ap-7678	451	2	0	0	NUM
ap-7678	451	3	dξ	dξ	PROPN
ap-7678	451	4	ξprl(ξ	ξprl(ξ	PROPN
ap-7678	451	5	)	)	PUNCT
ap-7678	451	6	or	or	CCONJ
ap-7678	451	7	wl(p	wl(p	PUNCT
ap-7678	451	8	)	)	PUNCT
ap-7678	451	9	=	=	NOUN
ap-7678	451	10	γ[p	γ[p	PROPN
ap-7678	451	11	+	+	CCONJ
ap-7678	451	12	1	1	NUM
ap-7678	451	13	2	2	NUM
ap-7678	451	14	]	]	PUNCT
ap-7678	451	15	satisfy	satisfy	VERB
ap-7678	451	16	the	the	DET
ap-7678	451	17	recursion	recursion	NOUN
ap-7678	451	18	relation	relation	NOUN
ap-7678	451	19	wl(p	wl(p	PUNCT
ap-7678	452	1	+	+	NOUN
ap-7678	452	2	1	1	X
ap-7678	452	3	)	)	PUNCT
ap-7678	452	4	=	=	NOUN
ap-7678	453	1	(	(	PUNCT
ap-7678	453	2	p	p	X
ap-7678	453	3	+	+	NOUN
ap-7678	453	4	1/2)wl(p	1/2)wl(p	NUM
ap-7678	453	5	)	)	PUNCT
ap-7678	453	6	;	;	PUNCT
ap-7678	453	7	whereas	whereas	SCONJ
ap-7678	453	8	wg(p	wg(p	NOUN
ap-7678	453	9	)	)	PUNCT
ap-7678	453	10	≡∫	≡∫	NOUN
ap-7678	453	11	∞	∞	PROPN
ap-7678	453	12	0	0	NUM
ap-7678	453	13	dξ	dξ	PROPN
ap-7678	453	14	ξprg(ξ	ξprg(ξ	NOUN
ap-7678	453	15	)	)	PUNCT
ap-7678	453	16	,	,	PUNCT
ap-7678	453	17	or	or	CCONJ
ap-7678	453	18	wg[p	wg[p	PROPN
ap-7678	453	19	]	]	X
ap-7678	454	1	=	=	SYM
ap-7678	454	2	2	2	NUM
ap-7678	454	3	2p−3	2p−3	NUM
ap-7678	454	4	4	4	NUM
ap-7678	454	5	γ	γ	X
ap-7678	454	6	[	[	X
ap-7678	454	7	2p+1	2p+1	NUM
ap-7678	454	8	4	4	NUM
ap-7678	454	9	]	]	PUNCT
ap-7678	454	10	,	,	PUNCT
ap-7678	454	11	satisfy	satisfy	VERB
ap-7678	454	12	the	the	DET
ap-7678	454	13	recursion	recursion	NOUN
ap-7678	454	14	relation	relation	NOUN
ap-7678	454	15	wg(p	wg(p	PUNCT
ap-7678	454	16	+	+	X
ap-7678	454	17	2	2	X
ap-7678	454	18	)	)	PUNCT
ap-7678	454	19	=	=	NOUN
ap-7678	455	1	(	(	PUNCT
ap-7678	455	2	p	p	X
ap-7678	455	3	+	+	NOUN
ap-7678	455	4	1/2)wg(p	1/2)wg(p	NUM
ap-7678	455	5	)	)	PUNCT
ap-7678	455	6	.	.	PUNCT
ap-7678	456	1	in	in	ADP
ap-7678	456	2	their	their	PRON
ap-7678	456	3	original	original	ADJ
ap-7678	456	4	work	work	NOUN
ap-7678	456	5	on	on	ADP
ap-7678	456	6	the	the	DET
ap-7678	456	7	oppq	oppq	NOUN
ap-7678	456	8	-	-	PUNCT
ap-7678	456	9	approximation	approximation	NOUN
ap-7678	456	10	method	method	NOUN
ap-7678	456	11	(	(	PUNCT
ap-7678	456	12	i.e.	i.e.	X
ap-7678	456	13	eq	eq	ADJ
ap-7678	456	14	.	.	PUNCT
ap-7678	456	15	(	(	PUNCT
ap-7678	456	16	28	28	NUM
ap-7678	456	17	)	)	PUNCT
ap-7678	456	18	)	)	PUNCT
ap-7678	456	19	,	,	PUNCT
ap-7678	456	20	handy	handy	ADJ
ap-7678	456	21	and	and	CCONJ
ap-7678	456	22	vrinceanu	vrinceanu	NOUN
ap-7678	456	23	implemented	implement	VERB
ap-7678	456	24	oppq	oppq	NOUN
ap-7678	456	25	-	-	PUNCT
ap-7678	456	26	am	be	AUX
ap-7678	456	27	with	with	ADP
ap-7678	456	28	both	both	PRON
ap-7678	456	29	of	of	ADP
ap-7678	456	30	these	these	DET
ap-7678	456	31	types	type	NOUN
ap-7678	456	32	of	of	ADP
ap-7678	456	33	weights	weight	NOUN
ap-7678	456	34	(	(	PUNCT
ap-7678	456	35	i.e.	i.e.	X
ap-7678	456	36	using	use	VERB
ap-7678	456	37	their	their	PRON
ap-7678	456	38	counterparts	counterpart	NOUN
ap-7678	456	39	along	along	ADP
ap-7678	456	40	the	the	DET
ap-7678	456	41	entire	entire	ADJ
ap-7678	456	42	real	real	ADJ
ap-7678	456	43	axis	axis	NOUN
ap-7678	456	44	,	,	PUNCT
ap-7678	456	45	x	x	X
ap-7678	456	46	∈	∈	PROPN
ap-7678	456	47	ℜ	ℜ	PROPN
ap-7678	456	48	)	)	PUNCT
ap-7678	456	49	.	.	PUNCT
ap-7678	457	1	they	they	PRON
ap-7678	457	2	showed	show	VERB
ap-7678	457	3	that	that	SCONJ
ap-7678	457	4	superior	superior	ADJ
ap-7678	457	5	(	(	PUNCT
ap-7678	457	6	faster	fast	ADV
ap-7678	457	7	converging	converge	VERB
ap-7678	457	8	)	)	PUNCT
ap-7678	457	9	results	result	NOUN
ap-7678	457	10	were	be	AUX
ap-7678	457	11	obtained	obtain	VERB
ap-7678	457	12	for	for	ADP
ap-7678	457	13	weights	weight	NOUN
ap-7678	457	14	that	that	PRON
ap-7678	457	15	emulated	emulate	VERB
ap-7678	457	16	the	the	DET
ap-7678	457	17	asymptotic	asymptotic	ADJ
ap-7678	457	18	form	form	NOUN
ap-7678	457	19	of	of	ADP
ap-7678	457	20	the	the	DET
ap-7678	457	21	physical	physical	ADJ
ap-7678	457	22	states	state	NOUN
ap-7678	457	23	.	.	PUNCT
ap-7678	458	1	the	the	DET
ap-7678	458	2	enhanced	enhance	VERB
ap-7678	458	3	efficiency	efficiency	NOUN
ap-7678	458	4	of	of	ADP
ap-7678	458	5	working	work	VERB
ap-7678	458	6	with	with	ADP
ap-7678	458	7	rg(ξ	rg(ξ	NOUN
ap-7678	458	8	)	)	PUNCT
ap-7678	458	9	instead	instead	ADV
ap-7678	458	10	of	of	ADP
ap-7678	458	11	rl(ξ	rl(ξ	NOUN
ap-7678	458	12	)	)	PUNCT
ap-7678	458	13	is	be	AUX
ap-7678	458	14	evident	evident	ADJ
ap-7678	458	15	in	in	ADP
ap-7678	458	16	figure	figure	NOUN
ap-7678	458	17	3	3	NUM
ap-7678	458	18	compared	compare	VERB
ap-7678	458	19	to	to	PART
ap-7678	458	20	figure	figure	VERB
ap-7678	458	21	2	2	NUM
ap-7678	458	22	(	(	PUNCT
ap-7678	458	23	i.e.	i.e.	X
ap-7678	458	24	the	the	DET
ap-7678	458	25	functions	function	NOUN
ap-7678	458	26	for	for	ADP
ap-7678	458	27	the	the	DET
ap-7678	458	28	former	former	ADJ
ap-7678	458	29	are	be	AUX
ap-7678	458	30	converging	converge	VERB
ap-7678	458	31	much	much	ADV
ap-7678	458	32	faster	fast	ADV
ap-7678	458	33	around	around	ADP
ap-7678	458	34	the	the	DET
ap-7678	458	35	physical	physical	ADJ
ap-7678	458	36	energies	energy	NOUN
ap-7678	458	37	)	)	PUNCT
ap-7678	458	38	.	.	PUNCT
ap-7678	459	1	the	the	DET
ap-7678	459	2	numerical	numerical	ADJ
ap-7678	459	3	results	result	NOUN
ap-7678	459	4	for	for	ADP
ap-7678	459	5	the	the	DET
ap-7678	459	6	ms	ms	PROPN
ap-7678	459	7	=	=	SYM
ap-7678	459	8	0	0	NUM
ap-7678	459	9	sextic	sextic	ADJ
ap-7678	459	10	anharmonic	anharmonic	ADJ
ap-7678	459	11	oscillator	oscillator	NOUN
ap-7678	459	12	problem	problem	NOUN
ap-7678	459	13	are	be	AUX
ap-7678	459	14	given	give	VERB
ap-7678	459	15	in	in	ADP
ap-7678	459	16	the	the	DET
ap-7678	459	17	following	follow	VERB
ap-7678	459	18	section	section	NOUN
ap-7678	459	19	.	.	PUNCT
ap-7678	460	1	we	we	PRON
ap-7678	460	2	only	only	ADV
ap-7678	460	3	give	give	VERB
ap-7678	460	4	numerical	numerical	ADJ
ap-7678	460	5	results	result	NOUN
ap-7678	460	6	based	base	VERB
ap-7678	460	7	on	on	ADP
ap-7678	460	8	using	use	VERB
ap-7678	460	9	rl(ξ	rl(ξ	NOUN
ap-7678	460	10	)	)	PUNCT
ap-7678	460	11	.	.	PUNCT
ap-7678	461	1	instead	instead	ADV
ap-7678	461	2	of	of	ADP
ap-7678	461	3	working	work	VERB
ap-7678	461	4	with	with	ADP
ap-7678	461	5	rg(ξ	rg(ξ	NOUN
ap-7678	461	6	)	)	PUNCT
ap-7678	461	7	,	,	PUNCT
ap-7678	461	8	we	we	PRON
ap-7678	461	9	will	will	AUX
ap-7678	461	10	use	use	VERB
ap-7678	461	11	the	the	DET
ap-7678	461	12	actual	actual	ADJ
ap-7678	461	13	ground	ground	NOUN
ap-7678	461	14	state	state	NOUN
ap-7678	461	15	wavefunction	wavefunction	NOUN
ap-7678	461	16	(	(	PUNCT
ap-7678	461	17	i.e.	i.e.	X
ap-7678	461	18	its	its	PRON
ap-7678	461	19	power	power	NOUN
ap-7678	461	20	moments	moment	NOUN
ap-7678	461	21	)	)	PUNCT
ap-7678	461	22	,	,	PUNCT
ap-7678	461	23	rgr(ξ	rgr(ξ	PROPN
ap-7678	461	24	)	)	PUNCT
ap-7678	461	25	.	.	PUNCT
ap-7678	462	1	the	the	DET
ap-7678	462	2	reason	reason	NOUN
ap-7678	462	3	is	be	AUX
ap-7678	462	4	that	that	SCONJ
ap-7678	462	5	both	both	PRON
ap-7678	462	6	have	have	VERB
ap-7678	462	7	the	the	DET
ap-7678	462	8	same	same	ADJ
ap-7678	462	9	asymptotic	asymptotic	ADJ
ap-7678	462	10	behavior	behavior	NOUN
ap-7678	462	11	.	.	PUNCT
ap-7678	463	1	that	that	PRON
ap-7678	463	2	is	be	AUX
ap-7678	463	3	,	,	PUNCT
ap-7678	463	4	rgr(ξ	rgr(ξ	PROPN
ap-7678	463	5	)	)	PUNCT
ap-7678	463	6	=	=	SYM
ap-7678	463	7	φgr(ξ	φgr(ξ	PROPN
ap-7678	463	8	)	)	PUNCT
ap-7678	463	9	∼	∼	NOUN
ap-7678	463	10	rg(ξ	rg(ξ	NOUN
ap-7678	463	11	)	)	PUNCT
ap-7678	463	12	;	;	PUNCT
ap-7678	463	13	therefore	therefore	ADV
ap-7678	463	14	,	,	PUNCT
ap-7678	463	15	upon	upon	SCONJ
ap-7678	463	16	solving	solve	VERB
ap-7678	463	17	for	for	ADP
ap-7678	463	18	the	the	DET
ap-7678	463	19	power	power	NOUN
ap-7678	463	20	moments	moment	NOUN
ap-7678	463	21	of	of	ADP
ap-7678	463	22	the	the	DET
ap-7678	463	23	ground	ground	NOUN
ap-7678	463	24	state	state	NOUN
ap-7678	463	25	wavefunction	wavefunction	NOUN
ap-7678	463	26	(	(	PUNCT
ap-7678	463	27	including	include	VERB
ap-7678	463	28	the	the	DET
ap-7678	463	29	energy	energy	NOUN
ap-7678	463	30	)	)	PUNCT
ap-7678	463	31	,	,	PUNCT
ap-7678	463	32	we	we	PRON
ap-7678	463	33	can	can	AUX
ap-7678	463	34	use	use	VERB
ap-7678	463	35	it	it	PRON
ap-7678	463	36	to	to	PART
ap-7678	463	37	generate	generate	VERB
ap-7678	463	38	its	its	PRON
ap-7678	463	39	orthonormal	orthonormal	ADJ
ap-7678	463	40	polynomials	polynomial	NOUN
ap-7678	463	41	,	,	PUNCT
ap-7678	463	42	and	and	CCONJ
ap-7678	463	43	quantize	quantize	VERB
ap-7678	463	44	the	the	DET
ap-7678	463	45	excited	excited	ADJ
ap-7678	463	46	states	state	NOUN
ap-7678	463	47	,	,	PUNCT
ap-7678	463	48	through	through	ADP
ap-7678	463	49	oppq	oppq	NOUN
ap-7678	463	50	.	.	PUNCT
ap-7678	464	1	to	to	PART
ap-7678	464	2	be	be	AUX
ap-7678	464	3	able	able	ADJ
ap-7678	464	4	to	to	PART
ap-7678	464	5	do	do	VERB
ap-7678	464	6	this	this	PRON
ap-7678	464	7	requires	require	VERB
ap-7678	464	8	sufficient	sufficient	ADJ
ap-7678	464	9	accuracy	accuracy	NOUN
ap-7678	464	10	in	in	ADP
ap-7678	464	11	the	the	DET
ap-7678	464	12	determination	determination	NOUN
ap-7678	464	13	of	of	ADP
ap-7678	464	14	the	the	DET
ap-7678	464	15	ground	ground	NOUN
ap-7678	464	16	state	state	NOUN
ap-7678	464	17	power	power	NOUN
ap-7678	464	18	moments	moment	NOUN
ap-7678	464	19	,	,	PUNCT
ap-7678	464	20	since	since	SCONJ
ap-7678	464	21	these	these	PRON
ap-7678	464	22	must	must	AUX
ap-7678	464	23	generate	generate	VERB
ap-7678	464	24	a	a	DET
ap-7678	464	25	positive	positive	ADJ
ap-7678	464	26	hankel	hankel	NOUN
ap-7678	464	27	matrix	matrix	NOUN
ap-7678	464	28	for	for	ADP
ap-7678	464	29	the	the	DET
ap-7678	464	30	weight	weight	NOUN
ap-7678	464	31	(	(	PUNCT
ap-7678	464	32	before	before	ADP
ap-7678	464	33	implementation	implementation	NOUN
ap-7678	464	34	of	of	ADP
ap-7678	464	35	a	a	DET
ap-7678	464	36	cholesky	cholesky	ADJ
ap-7678	464	37	analysis	analysis	NOUN
ap-7678	464	38	)	)	PUNCT
ap-7678	464	39	.	.	PUNCT
ap-7678	465	1	although	although	SCONJ
ap-7678	465	2	this	this	PRON
ap-7678	465	3	can	can	AUX
ap-7678	465	4	be	be	AUX
ap-7678	465	5	done	do	VERB
ap-7678	465	6	through	through	ADP
ap-7678	465	7	oppq	oppq	NOUN
ap-7678	465	8	,	,	PUNCT
ap-7678	465	9	emm	emm	PROPN
ap-7678	465	10	inherently	inherently	ADV
ap-7678	465	11	works	work	VERB
ap-7678	465	12	with	with	ADP
ap-7678	465	13	positive	positive	ADJ
ap-7678	465	14	matrices	matrix	NOUN
ap-7678	465	15	and	and	CCONJ
ap-7678	465	16	therefore	therefore	ADV
ap-7678	465	17	is	be	AUX
ap-7678	465	18	more	more	ADV
ap-7678	465	19	efficient	efficient	ADJ
ap-7678	465	20	for	for	ADP
ap-7678	465	21	doing	do	VERB
ap-7678	465	22	this	this	DET
ap-7678	465	23	type	type	NOUN
ap-7678	465	24	of	of	ADP
ap-7678	465	25	analysis	analysis	NOUN
ap-7678	465	26	.	.	PUNCT
ap-7678	466	1	9.5	9.5	NUM
ap-7678	466	2	.	.	PUNCT
ap-7678	467	1	oppq	oppq	PROPN
ap-7678	467	2	results	result	VERB
ap-7678	467	3	for	for	ADP
ap-7678	467	4	the	the	DET
ap-7678	467	5	ms	ms	PROPN
ap-7678	467	6	=	=	PROPN
ap-7678	467	7	0	0	PROPN
ap-7678	467	8	,	,	PUNCT
ap-7678	467	9	sextic	sextic	ADJ
ap-7678	467	10	anharmonic	anharmonic	ADJ
ap-7678	467	11	oscillator	oscillator	NOUN
ap-7678	467	12	the	the	DET
ap-7678	467	13	oppq	oppq	NOUN
ap-7678	467	14	-	-	PUNCT
ap-7678	467	15	φ	φ	PROPN
ap-7678	467	16	formulation	formulation	NOUN
ap-7678	467	17	is	be	AUX
ap-7678	467	18	a	a	DET
ap-7678	467	19	zero	zero	NUM
ap-7678	467	20	missing	missing	ADJ
ap-7678	467	21	moment	moment	NOUN
ap-7678	467	22	problem	problem	NOUN
ap-7678	467	23	(	(	PUNCT
ap-7678	467	24	i.e.	i.e.	X
ap-7678	467	25	ms	ms	X
ap-7678	467	26	=	=	PROPN
ap-7678	467	27	0	0	NUM
ap-7678	467	28	)	)	PUNCT
ap-7678	467	29	and	and	CCONJ
ap-7678	467	30	converges	converge	VERB
ap-7678	467	31	very	very	ADV
ap-7678	467	32	fast	fast	ADV
ap-7678	467	33	.	.	PUNCT
ap-7678	468	1	in	in	ADP
ap-7678	468	2	figure	figure	NOUN
ap-7678	468	3	2	2	NUM
ap-7678	468	4	we	we	PRON
ap-7678	468	5	plot	plot	VERB
ap-7678	468	6	the	the	DET
ap-7678	468	7	nested	nested	ADJ
ap-7678	468	8	sequence	sequence	NOUN
ap-7678	468	9	λn	λn	NOUN
ap-7678	468	10	(	(	PUNCT
ap-7678	468	11	e	e	NOUN
ap-7678	468	12	)	)	PUNCT
ap-7678	468	13	,	,	PUNCT
ap-7678	468	14	for	for	ADP
ap-7678	468	15	n	n	NOUN
ap-7678	468	16	=	=	SYM
ap-7678	468	17	4	4	NUM
ap-7678	468	18	,	,	PUNCT
ap-7678	468	19	6	6	NUM
ap-7678	468	20	,	,	PUNCT
ap-7678	468	21	8	8	NUM
ap-7678	468	22	,	,	PUNCT
ap-7678	468	23	10	10	NUM
ap-7678	468	24	,	,	PUNCT
ap-7678	468	25	12	12	NUM
ap-7678	468	26	.	.	PUNCT
ap-7678	469	1	this	this	DET
ap-7678	469	2	calculation	calculation	NOUN
ap-7678	469	3	was	be	AUX
ap-7678	469	4	done	do	VERB
ap-7678	469	5	based	base	VERB
ap-7678	469	6	on	on	ADP
ap-7678	469	7	the	the	DET
ap-7678	469	8	rl(ξ	rl(ξ	NOUN
ap-7678	469	9	)	)	PUNCT
ap-7678	469	10	weight	weight	NOUN
ap-7678	469	11	.	.	PUNCT
ap-7678	470	1	a	a	DET
ap-7678	470	2	similar	similar	ADJ
ap-7678	470	3	result	result	NOUN
ap-7678	470	4	holds	hold	VERB
ap-7678	470	5	for	for	ADP
ap-7678	470	6	the	the	DET
ap-7678	470	7	rg(ξ	rg(ξ	NOUN
ap-7678	470	8	)	)	PUNCT
ap-7678	470	9	weight	weight	NOUN
ap-7678	470	10	as	as	SCONJ
ap-7678	470	11	depicted	depict	VERB
ap-7678	470	12	in	in	ADP
ap-7678	470	13	figure	figure	NOUN
ap-7678	470	14	3	3	NUM
ap-7678	470	15	.	.	PUNCT
ap-7678	471	1	it	it	PRON
ap-7678	471	2	clearly	clearly	ADV
ap-7678	471	3	reveals	reveal	VERB
ap-7678	471	4	the	the	DET
ap-7678	471	5	faster	fast	ADJ
ap-7678	471	6	convergence	convergence	NOUN
ap-7678	471	7	afforded	afford	VERB
ap-7678	471	8	by	by	ADP
ap-7678	471	9	a	a	DET
ap-7678	471	10	weight	weight	NOUN
ap-7678	471	11	that	that	PRON
ap-7678	471	12	emulates	emulate	VERB
ap-7678	471	13	the	the	DET
ap-7678	471	14	asymptotic	asymptotic	ADJ
ap-7678	471	15	form	form	NOUN
ap-7678	471	16	of	of	ADP
ap-7678	471	17	the	the	DET
ap-7678	471	18	physical	physical	ADJ
ap-7678	471	19	states	state	NOUN
ap-7678	471	20	.	.	PUNCT
ap-7678	472	1	in	in	ADP
ap-7678	472	2	table	table	NOUN
ap-7678	472	3	5	5	NUM
ap-7678	472	4	we	we	PRON
ap-7678	472	5	give	give	VERB
ap-7678	472	6	the	the	DET
ap-7678	472	7	oppq	oppq	NOUN
ap-7678	472	8	-	-	PUNCT
ap-7678	472	9	bm	bm	PROPN
ap-7678	472	10	energy	energy	NOUN
ap-7678	472	11	estimates	estimate	NOUN
ap-7678	472	12	(	(	PUNCT
ap-7678	472	13	i.e.	i.e.	X
ap-7678	472	14	the	the	DET
ap-7678	472	15	local	local	ADJ
ap-7678	472	16	minima	minima	NOUN
ap-7678	472	17	,	,	PUNCT
ap-7678	472	18	∂eλn	∂eλn	PUNCT
ap-7678	472	19	(	(	PUNCT
ap-7678	472	20	e(min	e(min	PROPN
ap-7678	472	21	)	)	PUNCT
ap-7678	472	22	phys.;n	phys.;n	NOUN
ap-7678	472	23	)	)	PUNCT
ap-7678	473	1	=	=	PUNCT
ap-7678	473	2	0	0	X
ap-7678	473	3	)	)	PUNCT
ap-7678	473	4	for	for	ADP
ap-7678	473	5	the	the	DET
ap-7678	473	6	low	low	ADJ
ap-7678	473	7	lying	lie	VERB
ap-7678	473	8	,	,	PUNCT
ap-7678	473	9	even	even	ADV
ap-7678	473	10	parity	parity	NOUN
ap-7678	473	11	,	,	PUNCT
ap-7678	473	12	discrete	discrete	ADJ
ap-7678	473	13	state	state	NOUN
ap-7678	473	14	energies	energy	NOUN
ap-7678	473	15	,	,	PUNCT
ap-7678	473	16	based	base	VERB
ap-7678	473	17	on	on	ADP
ap-7678	473	18	eq	eq	ADP
ap-7678	473	19	.	.	PUNCT
ap-7678	474	1	(	(	PUNCT
ap-7678	474	2	65	65	NUM
ap-7678	474	3	)	)	PUNCT
ap-7678	474	4	,	,	PUNCT
ap-7678	474	5	using	use	VERB
ap-7678	474	6	the	the	DET
ap-7678	474	7	inferior	inferior	ADJ
ap-7678	474	8	weight	weight	NOUN
ap-7678	474	9	rl(ξ	rl(ξ	PUNCT
ap-7678	474	10	)	)	PUNCT
ap-7678	475	1	=	=	SYM
ap-7678	475	2	ξ−	ξ−	NOUN
ap-7678	475	3	1	1	NUM
ap-7678	475	4	2	2	NUM
ap-7678	475	5	exp(−ξ	exp(−ξ	PROPN
ap-7678	475	6	)	)	PUNCT
ap-7678	475	7	.	.	PUNCT
ap-7678	476	1	table	table	NOUN
ap-7678	476	2	5	5	NUM
ap-7678	476	3	only	only	ADV
ap-7678	476	4	cites	cite	VERB
ap-7678	476	5	results	result	NOUN
ap-7678	476	6	(	(	PUNCT
ap-7678	476	7	to	to	PART
ap-7678	476	8	ten	ten	NUM
ap-7678	476	9	significant	significant	ADJ
ap-7678	476	10	figures	figure	NOUN
ap-7678	476	11	)	)	PUNCT
ap-7678	476	12	for	for	ADP
ap-7678	476	13	n	n	DET
ap-7678	476	14	≤	≤	NUM
ap-7678	476	15	100	100	NUM
ap-7678	476	16	.	.	PUNCT
ap-7678	477	1	not	not	PART
ap-7678	477	2	shown	show	VERB
ap-7678	477	3	in	in	ADP
ap-7678	477	4	table	table	NOUN
ap-7678	477	5	5	5	NUM
ap-7678	477	6	are	be	AUX
ap-7678	477	7	the	the	DET
ap-7678	477	8	oppq	oppq	PROPN
ap-7678	477	9	-	-	PUNCT
ap-7678	477	10	bm	bm	PROPN
ap-7678	477	11	estimation	estimation	NOUN
ap-7678	477	12	results	result	NOUN
ap-7678	477	13	for	for	ADP
ap-7678	477	14	the	the	DET
ap-7678	477	15	ground	ground	NOUN
ap-7678	477	16	state	state	NOUN
ap-7678	477	17	at	at	ADP
ap-7678	477	18	n	n	NOUN
ap-7678	477	19	=	=	SYM
ap-7678	477	20	120	120	NUM
ap-7678	477	21	:	:	SYM
ap-7678	477	22	73	73	NUM
ap-7678	477	23	carlos	carlos	PROPN
ap-7678	477	24	r.	r.	PROPN
ap-7678	477	25	handy	handy	PROPN
ap-7678	477	26	acta	acta	PROPN
ap-7678	477	27	polytechnica	polytechnica	PROPN
ap-7678	477	28	n	n	ADP
ap-7678	477	29	∂esn	∂esn	ADV
ap-7678	477	30	(	(	PUNCT
ap-7678	477	31	e(min	e(min	PROPN
ap-7678	477	32	)	)	PUNCT
ap-7678	477	33	n	n	NOUN
ap-7678	477	34	)	)	PUNCT
ap-7678	477	35	=	=	SYM
ap-7678	477	36	0	0	PUNCT
ap-7678	477	37	log(sn	log(sn	ADJ
ap-7678	477	38	(	(	PUNCT
ap-7678	477	39	e(min	e(min	NOUN
ap-7678	477	40	)	)	PUNCT
ap-7678	477	41	n	n	NOUN
ap-7678	477	42	)	)	PUNCT
ap-7678	477	43	)	)	PUNCT
ap-7678	478	1	cn	cn	INTJ
ap-7678	478	2	(	(	PUNCT
ap-7678	478	3	en	en	X
ap-7678	478	4	)	)	PUNCT
ap-7678	478	5	=	=	SYM
ap-7678	478	6	0	0	NUM
ap-7678	478	7	e	e	X
ap-7678	478	8	(	(	PUNCT
ap-7678	478	9	l	l	NOUN
ap-7678	478	10	)	)	PUNCT
ap-7678	478	11	n	n	PRON
ap-7678	478	12	e	e	X
ap-7678	478	13	(	(	PUNCT
ap-7678	478	14	u	u	NOUN
ap-7678	478	15	)	)	PUNCT
ap-7678	478	16	n	n	ADV
ap-7678	478	17	20	20	NUM
ap-7678	478	18	49.94322409213197	49.94322409213197	NUM
ap-7678	478	19	7.982552803	7.982552803	NUM
ap-7678	478	20	49.4426432537	49.4426432537	NUM
ap-7678	478	21	48.893	48.893	NUM
ap-7678	478	22	50.857	50.857	NUM
ap-7678	478	23	30	30	NUM
ap-7678	478	24	49.45632020275004	49.45632020275004	NUM
ap-7678	478	25	7.997267757	7.997267757	NUM
ap-7678	478	26	49.4986057623	49.4986057623	NUM
ap-7678	478	27	49.366	49.366	NUM
ap-7678	478	28	49.546	49.546	NUM
ap-7678	478	29	40	40	NUM
ap-7678	478	30	49.51247497421839	49.51247497421839	NUM
ap-7678	478	31	8.002499541	8.002499541	NUM
ap-7678	478	32	49.5123070944	49.5123070944	NUM
ap-7678	478	33	49.504	49.504	NUM
ap-7678	478	34	49.520	49.520	NUM
ap-7678	478	35	50	50	NUM
ap-7678	478	36	49.51148774312169	49.51148774312169	NUM
ap-7678	478	37	8.002657778	8.002657778	NUM
ap-7678	478	38	49.5114731270	49.5114731270	NUM
ap-7678	478	39	49.510	49.510	NUM
ap-7678	478	40	49.512	49.512	NUM
ap-7678	478	41	60	60	NUM
ap-7678	478	42	49.51149945726214	49.51149945726214	NUM
ap-7678	478	43	8.002665930	8.002665930	NUM
ap-7678	478	44	49.5115007312	49.5115007312	NUM
ap-7678	478	45	49.51140	49.51140	NUM
ap-7678	478	46	49.51159	49.51159	NUM
ap-7678	478	47	70	70	NUM
ap-7678	478	48	49.51150043370514	49.51150043370514	NUM
ap-7678	478	49	8.002667120	8.002667120	NUM
ap-7678	478	50	49.5115003992	49.5115003992	NUM
ap-7678	478	51	49.51148	49.51148	NUM
ap-7678	478	52	49.51151	49.51151	NUM
ap-7678	478	53	80	80	NUM
ap-7678	478	54	49.51150037650008	49.51150037650008	NUM
ap-7678	478	55	8.002667337	8.002667337	NUM
ap-7678	478	56	49.511500375879294	49.511500375879294	NUM
ap-7678	478	57	49.5114990	49.5114990	NUM
ap-7678	478	58	49.5115017	49.5115017	NUM
ap-7678	478	59	90	90	NUM
ap-7678	478	60	49.51150037729806	49.51150037729806	NUM
ap-7678	478	61	8.002667346	8.002667346	NUM
ap-7678	478	62	49.511500377394135	49.511500377394135	NUM
ap-7678	478	63	49.5115001949	49.5115001949	NUM
ap-7678	478	64	49.5115005597	49.5115005597	NUM
ap-7678	478	65	100	100	NUM
ap-7678	478	66	49.51150037738496	49.51150037738496	NUM
ap-7678	478	67	8.002667348	8.002667348	NUM
ap-7678	478	68	49.511500377382972	49.511500377382972	NUM
ap-7678	478	69	49.5115003518	49.5115003518	NUM
ap-7678	478	70	49.5115004029	49.5115004029	NUM
ap-7678	478	71	110	110	NUM
ap-7678	478	72	49.51150037737999	49.51150037737999	NUM
ap-7678	478	73	8.002667348	8.002667348	NUM
ap-7678	478	74	49.511500377379789	49.511500377379789	NUM
ap-7678	478	75	49.5115003736	49.5115003736	NUM
ap-7678	478	76	49.5115003811	49.5115003811	NUM
ap-7678	478	77	120	120	NUM
ap-7678	478	78	49.51150037737991	49.51150037737991	NUM
ap-7678	478	79	8.002667348	8.002667348	NUM
ap-7678	478	80	49.511500377379918	49.511500377379918	NUM
ap-7678	478	81	49.5115003768	49.5115003768	NUM
ap-7678	478	82	49.5115003780	49.5115003780	NUM
ap-7678	478	83	b	b	NOUN
ap-7678	478	84	=	=	SYM
ap-7678	478	85	8.01	8.01	NUM
ap-7678	478	86	table	table	NOUN
ap-7678	478	87	6	6	NUM
ap-7678	478	88	.	.	PUNCT
ap-7678	479	1	oppq	oppq	PROPN
ap-7678	479	2	results	result	NOUN
ap-7678	479	3	for	for	ADP
ap-7678	479	4	e8	e8	PROPN
ap-7678	479	5	:	:	PUNCT
ap-7678	479	6	rl	rl	PROPN
ap-7678	479	7	=	=	SYM
ap-7678	479	8	e−ξ√	e−ξ√	PROPN
ap-7678	479	9	ξ	ξ	PROPN
ap-7678	479	10	.	.	PUNCT
ap-7678	480	1	egr	egr	PROPN
ap-7678	480	2	=	=	PROPN
ap-7678	480	3	−.52326862212755224	−.52326862212755224	PROPN
ap-7678	480	4	.	.	PUNCT
ap-7678	481	1	this	this	DET
ap-7678	481	2	result	result	NOUN
ap-7678	481	3	lies	lie	VERB
ap-7678	481	4	within	within	ADP
ap-7678	481	5	the	the	DET
ap-7678	481	6	bounds	bound	NOUN
ap-7678	481	7	generated	generate	VERB
ap-7678	481	8	through	through	ADP
ap-7678	481	9	an	an	DET
ap-7678	481	10	emm	emm	PROPN
ap-7678	481	11	-	-	PUNCT
ap-7678	481	12	φ	φ	PROPN
ap-7678	481	13	analysis	analysis	NOUN
ap-7678	481	14	(	(	PUNCT
ap-7678	481	15	based	base	VERB
ap-7678	481	16	on	on	ADP
ap-7678	481	17	the	the	DET
ap-7678	481	18	first	first	ADJ
ap-7678	481	19	29	29	NUM
ap-7678	481	20	power	power	NOUN
ap-7678	481	21	moments	moment	NOUN
ap-7678	481	22	):	):	PUNCT
ap-7678	481	23	−.5232686221275616	−.5232686221275616	X
ap-7678	481	24	<	<	X
ap-7678	481	25	egr	egr	X
ap-7678	481	26	<	<	X
ap-7678	481	27	−.5232686221275495	−.5232686221275495	PROPN
ap-7678	481	28	.	.	PUNCT
ap-7678	482	1	(	(	PUNCT
ap-7678	482	2	68	68	NUM
ap-7678	482	3	)	)	PUNCT
ap-7678	482	4	we	we	PRON
ap-7678	482	5	can	can	AUX
ap-7678	482	6	infer	infer	VERB
ap-7678	482	7	that	that	SCONJ
ap-7678	482	8	the	the	DET
ap-7678	482	9	emm	emm	PROPN
ap-7678	482	10	bounds	bound	VERB
ap-7678	482	11	in	in	ADP
ap-7678	482	12	eq	eq	ADP
ap-7678	482	13	.	.	PUNCT
ap-7678	483	1	(	(	PUNCT
ap-7678	483	2	68	68	NUM
ap-7678	483	3	)	)	PUNCT
ap-7678	483	4	predict	predict	VERB
ap-7678	483	5	the	the	DET
ap-7678	483	6	exact	exact	ADJ
ap-7678	483	7	answer	answer	NOUN
ap-7678	483	8	to	to	ADP
ap-7678	483	9	13	13	NUM
ap-7678	483	10	decimal	decimal	ADJ
ap-7678	483	11	places	place	NOUN
ap-7678	483	12	.	.	PUNCT
ap-7678	484	1	this	this	DET
ap-7678	484	2	result	result	NOUN
ap-7678	484	3	,	,	PUNCT
ap-7678	484	4	based	base	VERB
ap-7678	484	5	on	on	ADP
ap-7678	484	6	29	29	NUM
ap-7678	484	7	moments	moment	NOUN
ap-7678	484	8	,	,	PUNCT
ap-7678	484	9	surpasses	surpass	VERB
ap-7678	484	10	the	the	DET
ap-7678	484	11	oppq	oppq	PROPN
ap-7678	484	12	-	-	PUNCT
ap-7678	484	13	bm	bm	PROPN
ap-7678	484	14	estimation	estimation	NOUN
ap-7678	484	15	analysis	analysis	NOUN
ap-7678	484	16	in	in	ADP
ap-7678	484	17	table	table	NOUN
ap-7678	484	18	5	5	NUM
ap-7678	484	19	(	(	PUNCT
ap-7678	484	20	n	n	NOUN
ap-7678	484	21	=	=	SYM
ap-7678	484	22	30	30	NUM
ap-7678	484	23	)	)	PUNCT
ap-7678	484	24	of	of	ADP
ap-7678	484	25	−0.5232712343	−0.5232712343	PROPN
ap-7678	484	26	,	,	PUNCT
ap-7678	484	27	at	at	ADP
ap-7678	484	28	the	the	DET
ap-7678	484	29	fourth	fourth	ADJ
ap-7678	484	30	-	-	PUNCT
ap-7678	484	31	fifth	fifth	ADJ
ap-7678	484	32	decimal	decimal	ADJ
ap-7678	484	33	place	place	NOUN
ap-7678	484	34	.	.	PUNCT
ap-7678	485	1	thus	thus	ADV
ap-7678	485	2	emm	emm	PROPN
ap-7678	485	3	-	-	PUNCT
ap-7678	485	4	φ	φ	PROPN
ap-7678	485	5	,	,	PUNCT
ap-7678	485	6	by	by	ADP
ap-7678	485	7	this	this	DET
ap-7678	485	8	comparison	comparison	NOUN
ap-7678	485	9	,	,	PUNCT
ap-7678	485	10	is	be	AUX
ap-7678	485	11	approximately	approximately	ADV
ap-7678	485	12	three	three	NUM
ap-7678	485	13	times	time	NOUN
ap-7678	485	14	more	more	ADV
ap-7678	485	15	efficient	efficient	ADJ
ap-7678	485	16	.	.	PUNCT
ap-7678	486	1	having	having	AUX
ap-7678	486	2	said	say	VERB
ap-7678	486	3	this	this	PRON
ap-7678	486	4	,	,	PUNCT
ap-7678	486	5	we	we	PRON
ap-7678	486	6	remind	remind	VERB
ap-7678	486	7	the	the	DET
ap-7678	486	8	reader	reader	NOUN
ap-7678	486	9	that	that	SCONJ
ap-7678	486	10	whereas	whereas	SCONJ
ap-7678	486	11	emm	emm	PROPN
ap-7678	486	12	involves	involve	VERB
ap-7678	486	13	sophisticated	sophisticated	ADJ
ap-7678	486	14	analysis	analysis	NOUN
ap-7678	486	15	(	(	PUNCT
ap-7678	486	16	i.e.	i.e.	X
ap-7678	486	17	nonlinear	nonlinear	ADJ
ap-7678	486	18	convex	convex	NOUN
ap-7678	486	19	optimization	optimization	NOUN
ap-7678	486	20	)	)	PUNCT
ap-7678	486	21	,	,	PUNCT
ap-7678	486	22	oppq	oppq	PROPN
ap-7678	486	23	is	be	AUX
ap-7678	486	24	purely	purely	ADV
ap-7678	486	25	algebraic	algebraic	ADJ
ap-7678	486	26	,	,	PUNCT
ap-7678	486	27	and	and	CCONJ
ap-7678	486	28	implementable	implementable	ADJ
ap-7678	486	29	to	to	ADP
ap-7678	486	30	arbitrary	arbitrary	ADJ
ap-7678	486	31	accuracy	accuracy	NOUN
ap-7678	486	32	through	through	ADP
ap-7678	486	33	algebraic	algebraic	ADJ
ap-7678	486	34	software	software	NOUN
ap-7678	486	35	,	,	PUNCT
ap-7678	486	36	such	such	ADJ
ap-7678	486	37	as	as	ADP
ap-7678	486	38	mathematica	mathematica	PROPN
ap-7678	486	39	.	.	PUNCT
ap-7678	487	1	additionally	additionally	ADV
ap-7678	487	2	,	,	PUNCT
ap-7678	487	3	emm	emm	PROPN
ap-7678	487	4	is	be	AUX
ap-7678	487	5	,	,	PUNCT
ap-7678	487	6	in	in	ADP
ap-7678	487	7	practice	practice	NOUN
ap-7678	487	8	,	,	PUNCT
ap-7678	487	9	only	only	ADV
ap-7678	487	10	applicable	applicable	ADJ
ap-7678	487	11	to	to	ADP
ap-7678	487	12	the	the	DET
ap-7678	487	13	(	(	PUNCT
ap-7678	487	14	multidimensional	multidimensional	ADJ
ap-7678	487	15	)	)	PUNCT
ap-7678	487	16	bosonic	bosonic	PROPN
ap-7678	487	17	ground	ground	NOUN
ap-7678	487	18	state	state	NOUN
ap-7678	487	19	;	;	PUNCT
ap-7678	487	20	whereas	whereas	SCONJ
ap-7678	487	21	oppq	oppq	PROPN
ap-7678	487	22	applies	apply	VERB
ap-7678	487	23	to	to	ADP
ap-7678	487	24	any	any	DET
ap-7678	487	25	multidimensional	multidimensional	ADJ
ap-7678	487	26	bosonic	bosonic	NOUN
ap-7678	487	27	or	or	CCONJ
ap-7678	487	28	fermionic	fermionic	NOUN
ap-7678	487	29	(	(	PUNCT
ap-7678	487	30	low	low	ADJ
ap-7678	487	31	dimension	dimension	NOUN
ap-7678	487	32	)	)	PUNCT
ap-7678	487	33	system	system	NOUN
ap-7678	487	34	,	,	PUNCT
ap-7678	487	35	including	include	VERB
ap-7678	487	36	non	non	ADJ
ap-7678	487	37	-	-	ADJ
ap-7678	487	38	hermitian	hermitian	ADJ
ap-7678	487	39	systems	system	NOUN
ap-7678	487	40	.	.	PUNCT
ap-7678	488	1	9.5.1	9.5.1	X
ap-7678	488	2	.	.	PUNCT
ap-7678	488	3	generating	generate	VERB
ap-7678	488	4	bounds	bound	NOUN
ap-7678	488	5	for	for	ADP
ap-7678	488	6	e8	e8	PROPN
ap-7678	488	7	we	we	PRON
ap-7678	488	8	re	re	VERB
ap-7678	488	9	-	-	VERB
ap-7678	488	10	examine	examine	VERB
ap-7678	488	11	the	the	DET
ap-7678	488	12	results	result	NOUN
ap-7678	488	13	in	in	ADP
ap-7678	488	14	table	table	NOUN
ap-7678	488	15	5	5	NUM
ap-7678	488	16	for	for	ADP
ap-7678	488	17	e8	e8	PROPN
ap-7678	488	18	,	,	PUNCT
ap-7678	488	19	the	the	DET
ap-7678	488	20	slowest	slow	ADJ
ap-7678	488	21	converging	converge	VERB
ap-7678	488	22	energy	energy	NOUN
ap-7678	488	23	.	.	PUNCT
ap-7678	489	1	as	as	ADP
ap-7678	489	2	with	with	ADP
ap-7678	489	3	other	other	ADJ
ap-7678	489	4	methods	method	NOUN
ap-7678	489	5	,	,	PUNCT
ap-7678	489	6	such	such	ADJ
ap-7678	489	7	as	as	ADP
ap-7678	489	8	rayleigh	rayleigh	PROPN
ap-7678	489	9	-	-	PUNCT
ap-7678	489	10	ritz	ritz	PROPN
ap-7678	489	11	(	(	PUNCT
ap-7678	489	12	rr	rr	PROPN
ap-7678	489	13	)	)	PUNCT
ap-7678	489	14	,	,	PUNCT
ap-7678	489	15	the	the	DET
ap-7678	489	16	manifest	manifest	ADJ
ap-7678	489	17	convergence	convergence	NOUN
ap-7678	489	18	of	of	ADP
ap-7678	489	19	the	the	DET
ap-7678	489	20	results	result	NOUN
ap-7678	489	21	(	(	PUNCT
ap-7678	489	22	i.e.	i.e.	X
ap-7678	489	23	the	the	DET
ap-7678	489	24	n	n	NOUN
ap-7678	489	25	→	→	SYM
ap-7678	489	26	120	120	NUM
ap-7678	489	27	sequence	sequence	NOUN
ap-7678	489	28	in	in	ADP
ap-7678	489	29	table	table	NOUN
ap-7678	489	30	5	5	NUM
ap-7678	489	31	)	)	PUNCT
ap-7678	489	32	is	be	AUX
ap-7678	489	33	no	no	DET
ap-7678	489	34	guarantee	guarantee	NOUN
ap-7678	489	35	of	of	ADP
ap-7678	489	36	the	the	DET
ap-7678	489	37	accuracy	accuracy	NOUN
ap-7678	489	38	of	of	ADP
ap-7678	489	39	the	the	DET
ap-7678	489	40	apparent	apparent	ADJ
ap-7678	489	41	limit	limit	NOUN
ap-7678	489	42	in	in	ADP
ap-7678	489	43	predicting	predict	VERB
ap-7678	489	44	the	the	DET
ap-7678	489	45	true	true	ADJ
ap-7678	489	46	energy	energy	NOUN
ap-7678	489	47	.	.	PUNCT
ap-7678	490	1	whereas	whereas	SCONJ
ap-7678	490	2	emm	emm	PROPN
ap-7678	490	3	produces	produce	VERB
ap-7678	490	4	converging	converge	VERB
ap-7678	490	5	bounds	bound	NOUN
ap-7678	490	6	from	from	ADP
ap-7678	490	7	first	first	ADJ
ap-7678	490	8	principles	principle	NOUN
ap-7678	490	9	;	;	PUNCT
ap-7678	490	10	oppq	oppq	PROPN
ap-7678	490	11	defines	define	VERB
ap-7678	490	12	a	a	DET
ap-7678	490	13	procedure	procedure	NOUN
ap-7678	490	14	by	by	ADP
ap-7678	490	15	which	which	PRON
ap-7678	490	16	one	one	PRON
ap-7678	490	17	can	can	AUX
ap-7678	490	18	generate	generate	VERB
ap-7678	490	19	bounds	bound	NOUN
ap-7678	490	20	provided	provide	VERB
ap-7678	490	21	a	a	DET
ap-7678	490	22	certain	certain	ADJ
ap-7678	490	23	parameter	parameter	NOUN
ap-7678	490	24	is	be	AUX
ap-7678	490	25	empirically	empirically	ADV
ap-7678	490	26	determined	determine	VERB
ap-7678	490	27	,	,	PUNCT
ap-7678	490	28	specifically	specifically	ADV
ap-7678	490	29	the	the	DET
ap-7678	490	30	coarse	coarse	ADJ
ap-7678	490	31	upper	upper	ADJ
ap-7678	490	32	bound	bind	VERB
ap-7678	490	33	,	,	PUNCT
ap-7678	490	34	b(u	b(u	PROPN
ap-7678	490	35	)	)	PUNCT
ap-7678	490	36	.	.	PUNCT
ap-7678	491	1	below	below	ADP
ap-7678	491	2	we	we	PRON
ap-7678	491	3	describe	describe	VERB
ap-7678	491	4	the	the	DET
ap-7678	491	5	oppq	oppq	NOUN
ap-7678	491	6	-	-	PUNCT
ap-7678	491	7	bm	bm	PROPN
ap-7678	491	8	bounding	bounding	NOUN
ap-7678	491	9	procedure	procedure	NOUN
ap-7678	491	10	in	in	ADP
ap-7678	491	11	detail	detail	NOUN
ap-7678	491	12	,	,	PUNCT
ap-7678	491	13	although	although	SCONJ
ap-7678	491	14	the	the	DET
ap-7678	491	15	same	same	ADJ
ap-7678	491	16	underlies	underlie	VERB
ap-7678	491	17	the	the	DET
ap-7678	491	18	results	result	NOUN
ap-7678	491	19	in	in	ADP
ap-7678	491	20	tables	table	NOUN
ap-7678	491	21	4	4	NUM
ap-7678	491	22	and	and	CCONJ
ap-7678	491	23	3	3	NUM
ap-7678	491	24	,	,	PUNCT
ap-7678	491	25	for	for	ADP
ap-7678	491	26	the	the	DET
ap-7678	491	27	ms	ms	PROPN
ap-7678	491	28	=	=	PROPN
ap-7678	491	29	3	3	NUM
ap-7678	491	30	mer	mer	NOUN
ap-7678	491	31	formulation	formulation	NOUN
ap-7678	491	32	in	in	ADP
ap-7678	491	33	eq	eq	ADP
ap-7678	491	34	.	.	PUNCT
ap-7678	492	1	(	(	PUNCT
ap-7678	492	2	48	48	NUM
ap-7678	492	3	)	)	PUNCT
ap-7678	492	4	.	.	PUNCT
ap-7678	493	1	the	the	DET
ap-7678	493	2	only	only	ADJ
ap-7678	493	3	advantage	advantage	NOUN
ap-7678	493	4	of	of	ADP
ap-7678	493	5	the	the	DET
ap-7678	493	6	current	current	ADJ
ap-7678	493	7	problem	problem	NOUN
ap-7678	493	8	is	be	AUX
ap-7678	493	9	that	that	SCONJ
ap-7678	493	10	it	it	PRON
ap-7678	493	11	is	be	AUX
ap-7678	493	12	a	a	DET
ap-7678	493	13	zero	zero	NUM
ap-7678	493	14	missing	missing	ADJ
ap-7678	493	15	moment	moment	NOUN
ap-7678	493	16	problem	problem	NOUN
ap-7678	493	17	,	,	PUNCT
ap-7678	493	18	and	and	CCONJ
ap-7678	493	19	therefore	therefore	ADV
ap-7678	493	20	easier	easy	ADJ
ap-7678	493	21	to	to	PART
ap-7678	493	22	implement	implement	VERB
ap-7678	493	23	.	.	PUNCT
ap-7678	494	1	in	in	ADP
ap-7678	494	2	table	table	NOUN
ap-7678	494	3	6	6	NUM
ap-7678	494	4	we	we	PRON
ap-7678	494	5	provide	provide	VERB
ap-7678	494	6	the	the	DET
ap-7678	494	7	oppq	oppq	NOUN
ap-7678	494	8	-	-	PUNCT
ap-7678	494	9	bm	bm	PROPN
ap-7678	494	10	eigenenergy	eigenenergy	PROPN
ap-7678	494	11	estimate	estimate	NOUN
ap-7678	494	12	for	for	ADP
ap-7678	494	13	e8	e8	PROPN
ap-7678	494	14	(	(	PUNCT
ap-7678	494	15	the	the	DET
ap-7678	494	16	second	second	ADJ
ap-7678	494	17	column	column	NOUN
ap-7678	494	18	)	)	PUNCT
ap-7678	494	19	.	.	PUNCT
ap-7678	495	1	the	the	DET
ap-7678	495	2	third	third	ADJ
ap-7678	495	3	column	column	NOUN
ap-7678	495	4	contains	contain	VERB
ap-7678	495	5	the	the	DET
ap-7678	495	6	increasing	increase	VERB
ap-7678	495	7	,	,	PUNCT
ap-7678	495	8	convergent	convergent	NOUN
ap-7678	495	9	,	,	PUNCT
ap-7678	495	10	positive	positive	ADJ
ap-7678	495	11	sequence	sequence	NOUN
ap-7678	495	12	from	from	ADP
ap-7678	495	13	which	which	PRON
ap-7678	495	14	a	a	DET
ap-7678	495	15	coarse	coarse	ADJ
ap-7678	495	16	upper	upper	ADJ
ap-7678	495	17	bound	bind	VERB
ap-7678	495	18	,	,	PUNCT
ap-7678	495	19	b(u	b(u	PROPN
ap-7678	495	20	)	)	PUNCT
ap-7678	495	21	,	,	PUNCT
ap-7678	495	22	is	be	AUX
ap-7678	495	23	to	to	PART
ap-7678	495	24	be	be	AUX
ap-7678	495	25	empirically	empirically	ADV
ap-7678	495	26	determined	determine	VERB
ap-7678	495	27	.	.	PUNCT
ap-7678	496	1	using	use	VERB
ap-7678	496	2	this	this	DET
ap-7678	496	3	coarse	coarse	ADJ
ap-7678	496	4	upper	upper	ADJ
ap-7678	496	5	bound	bind	VERB
ap-7678	496	6	,	,	PUNCT
ap-7678	496	7	we	we	PRON
ap-7678	496	8	can	can	AUX
ap-7678	496	9	generate	generate	VERB
ap-7678	496	10	arbitrarily	arbitrarily	ADV
ap-7678	496	11	tight	tight	ADJ
ap-7678	496	12	bounds	bound	NOUN
ap-7678	496	13	(	(	PUNCT
ap-7678	496	14	i.e.	i.e.	X
ap-7678	496	15	the	the	DET
ap-7678	496	16	last	last	ADJ
ap-7678	496	17	two	two	NUM
ap-7678	496	18	columns	column	NOUN
ap-7678	496	19	)	)	PUNCT
ap-7678	496	20	,	,	PUNCT
ap-7678	496	21	as	as	SCONJ
ap-7678	496	22	n	n	PROPN
ap-7678	496	23	→	→	SYM
ap-7678	496	24	∞.	∞.	PROPN
ap-7678	496	25	note	note	VERB
ap-7678	496	26	that	that	SCONJ
ap-7678	496	27	we	we	PRON
ap-7678	496	28	continued	continue	VERB
ap-7678	496	29	generating	generate	VERB
ap-7678	496	30	these	these	DET
ap-7678	496	31	bounds	bound	NOUN
ap-7678	496	32	in	in	ADP
ap-7678	496	33	table	table	NOUN
ap-7678	496	34	7	7	NUM
ap-7678	496	35	.	.	PUNCT
ap-7678	497	1	this	this	DET
ap-7678	497	2	entire	entire	ADJ
ap-7678	497	3	procedure	procedure	NOUN
ap-7678	497	4	is	be	AUX
ap-7678	497	5	based	base	VERB
ap-7678	497	6	on	on	ADP
ap-7678	497	7	the	the	DET
ap-7678	497	8	assumption	assumption	NOUN
ap-7678	497	9	that	that	SCONJ
ap-7678	497	10	the	the	DET
ap-7678	497	11	manifest	manif	ADJ
ap-7678	497	12	convergence	convergence	NOUN
ap-7678	497	13	of	of	ADP
ap-7678	497	14	the	the	DET
ap-7678	497	15	third	third	ADJ
ap-7678	497	16	column	column	NOUN
ap-7678	497	17	in	in	ADP
ap-7678	497	18	table	table	NOUN
ap-7678	497	19	6	6	NUM
ap-7678	497	20	is	be	AUX
ap-7678	497	21	correctly	correctly	ADV
ap-7678	497	22	bounded	bound	VERB
ap-7678	497	23	,	,	PUNCT
ap-7678	497	24	from	from	ADP
ap-7678	497	25	above	above	ADV
ap-7678	497	26	,	,	PUNCT
ap-7678	497	27	by	by	ADP
ap-7678	497	28	the	the	DET
ap-7678	497	29	empirically	empirically	ADV
ap-7678	497	30	determined	determine	VERB
ap-7678	497	31	coarse	coarse	ADJ
ap-7678	497	32	upper	upper	ADJ
ap-7678	497	33	bound	bind	VERB
ap-7678	497	34	,	,	PUNCT
ap-7678	497	35	b(u	b(u	PROPN
ap-7678	497	36	)	)	PUNCT
ap-7678	497	37	.	.	PUNCT
ap-7678	498	1	the	the	DET
ap-7678	498	2	fourth	fourth	ADJ
ap-7678	498	3	column	column	NOUN
ap-7678	498	4	contains	contain	VERB
ap-7678	498	5	the	the	DET
ap-7678	498	6	oppq	oppq	NOUN
ap-7678	498	7	-	-	PUNCT
ap-7678	498	8	am	am	NOUN
ap-7678	498	9	estimate	estimate	NOUN
ap-7678	498	10	.	.	PUNCT
ap-7678	499	1	it	it	PRON
ap-7678	499	2	is	be	AUX
ap-7678	499	3	important	important	ADJ
ap-7678	499	4	to	to	PART
ap-7678	499	5	appreciate	appreciate	VERB
ap-7678	499	6	that	that	SCONJ
ap-7678	499	7	the	the	DET
ap-7678	499	8	coarseness	coarseness	NOUN
ap-7678	499	9	of	of	ADP
ap-7678	499	10	b(u	b(u	PROPN
ap-7678	499	11	)	)	PUNCT
ap-7678	499	12	has	have	VERB
ap-7678	499	13	nothing	nothing	PRON
ap-7678	499	14	to	to	PART
ap-7678	499	15	do	do	VERB
ap-7678	499	16	with	with	ADP
ap-7678	499	17	the	the	DET
ap-7678	499	18	tightness	tightness	NOUN
ap-7678	499	19	of	of	ADP
ap-7678	499	20	the	the	DET
ap-7678	499	21	bounds	bound	NOUN
ap-7678	499	22	,	,	PUNCT
ap-7678	499	23	in	in	ADP
ap-7678	499	24	principle	principle	NOUN
ap-7678	499	25	,	,	PUNCT
ap-7678	499	26	assuming	assume	VERB
ap-7678	499	27	one	one	PRON
ap-7678	499	28	can	can	AUX
ap-7678	499	29	generate	generate	VERB
ap-7678	499	30	high	high	ADJ
ap-7678	499	31	oppq	oppq	ADJ
ap-7678	499	32	expansion	expansion	NOUN
ap-7678	499	33	orders	order	NOUN
ap-7678	499	34	.	.	PUNCT
ap-7678	500	1	below	below	ADP
ap-7678	500	2	we	we	PRON
ap-7678	500	3	compare	compare	VERB
ap-7678	500	4	the	the	DET
ap-7678	500	5	accuracy	accuracy	NOUN
ap-7678	500	6	of	of	ADP
ap-7678	500	7	the	the	DET
ap-7678	500	8	oppqbm	oppqbm	NOUN
ap-7678	500	9	formalism	formalism	NOUN
ap-7678	500	10	to	to	PART
ap-7678	500	11	order	order	VERB
ap-7678	500	12	n	n	PRON
ap-7678	500	13	−	−	PROPN
ap-7678	500	14	120	120	NUM
ap-7678	500	15	,	,	PUNCT
ap-7678	500	16	both	both	CCONJ
ap-7678	500	17	with	with	ADP
ap-7678	500	18	respect	respect	NOUN
ap-7678	500	19	to	to	ADP
ap-7678	500	20	the	the	DET
ap-7678	500	21	oppq	oppq	PROPN
ap-7678	500	22	-	-	PUNCT
ap-7678	500	23	bm	bm	PROPN
ap-7678	500	24	energy	energy	NOUN
ap-7678	500	25	estimate	estimate	NOUN
ap-7678	500	26	(	(	PUNCT
ap-7678	500	27	e8	e8	PROPN
ap-7678	500	28	=	=	SYM
ap-7678	500	29	49.5115003774	49.5115003774	NUM
ap-7678	500	30	,	,	PUNCT
ap-7678	500	31	from	from	ADP
ap-7678	500	32	table	table	NOUN
ap-7678	500	33	6	6	NUM
ap-7678	500	34	)	)	PUNCT
ap-7678	500	35	and	and	CCONJ
ap-7678	500	36	the	the	DET
ap-7678	500	37	oppq	oppq	PROPN
ap-7678	500	38	-	-	PUNCT
ap-7678	500	39	bm	bm	PROPN
ap-7678	500	40	bounds	bound	NOUN
ap-7678	500	41	(	(	PUNCT
ap-7678	500	42	49.5115003768	49.5115003768	NUM
ap-7678	500	43	<	<	X
ap-7678	500	44	e8	e8	PROPN
ap-7678	500	45	<	<	X
ap-7678	500	46	49.5115003780	49.5115003780	NUM
ap-7678	500	47	,	,	PUNCT
ap-7678	500	48	also	also	ADV
ap-7678	500	49	from	from	ADP
ap-7678	500	50	table	table	NOUN
ap-7678	500	51	6	6	NUM
ap-7678	500	52	)	)	PUNCT
ap-7678	500	53	,	,	PUNCT
ap-7678	500	54	as	as	SCONJ
ap-7678	500	55	compared	compare	VERB
ap-7678	500	56	to	to	ADP
ap-7678	500	57	the	the	DET
ap-7678	500	58	bounds	bound	NOUN
ap-7678	500	59	generated	generate	VERB
ap-7678	500	60	by	by	ADP
ap-7678	500	61	an	an	DET
ap-7678	500	62	emm	emm	PROPN
ap-7678	500	63	analysis	analysis	NOUN
ap-7678	500	64	.	.	PUNCT
ap-7678	501	1	the	the	DET
ap-7678	501	2	results	result	NOUN
ap-7678	501	3	are	be	AUX
ap-7678	501	4	very	very	ADV
ap-7678	501	5	good	good	ADJ
ap-7678	501	6	and	and	CCONJ
ap-7678	501	7	consistent	consistent	ADJ
ap-7678	501	8	.	.	PUNCT
ap-7678	502	1	it	it	PRON
ap-7678	502	2	is	be	AUX
ap-7678	502	3	important	important	ADJ
ap-7678	502	4	to	to	PART
ap-7678	502	5	note	note	VERB
ap-7678	502	6	that	that	SCONJ
ap-7678	502	7	the	the	DET
ap-7678	502	8	oppq	oppq	PROPN
ap-7678	502	9	-	-	PUNCT
ap-7678	502	10	bm	bm	PROPN
ap-7678	502	11	estimate	estimate	NOUN
ap-7678	502	12	obtained	obtain	VERB
ap-7678	502	13	at	at	ADP
ap-7678	502	14	lower	low	ADJ
ap-7678	502	15	order	order	NOUN
ap-7678	502	16	,	,	PUNCT
ap-7678	502	17	does	do	AUX
ap-7678	502	18	not	not	PART
ap-7678	502	19	have	have	VERB
ap-7678	502	20	to	to	PART
ap-7678	502	21	lie	lie	VERB
ap-7678	502	22	within	within	ADP
ap-7678	502	23	the	the	DET
ap-7678	502	24	oppq	oppq	PROPN
ap-7678	502	25	-	-	PUNCT
ap-7678	502	26	bm	bm	PROPN
ap-7678	502	27	bounds	bound	NOUN
ap-7678	502	28	obtained	obtain	VERB
ap-7678	502	29	at	at	ADP
ap-7678	502	30	higher	high	ADJ
ap-7678	502	31	order	order	NOUN
ap-7678	502	32	.	.	PUNCT
ap-7678	503	1	thus	thus	ADV
ap-7678	503	2	the	the	DET
ap-7678	503	3	entry	entry	NOUN
ap-7678	503	4	in	in	ADP
ap-7678	503	5	the	the	DET
ap-7678	503	6	second	second	ADJ
ap-7678	503	7	column	column	NOUN
ap-7678	503	8	in	in	ADP
ap-7678	503	9	table	table	NOUN
ap-7678	503	10	6	6	NUM
ap-7678	503	11	,	,	PUNCT
ap-7678	503	12	corresponding	correspond	VERB
ap-7678	503	13	to	to	ADP
ap-7678	503	14	oppq	oppq	PROPN
ap-7678	503	15	-	-	PUNCT
ap-7678	503	16	bm	bm	PROPN
ap-7678	503	17	energy	energy	NOUN
ap-7678	503	18	estimate	estimate	NOUN
ap-7678	503	19	(	(	PUNCT
ap-7678	503	20	i.e.	i.e.	X
ap-7678	503	21	the	the	DET
ap-7678	503	22	local	local	ADJ
ap-7678	503	23	minima	minima	NOUN
ap-7678	503	24	)	)	PUNCT
ap-7678	503	25	of	of	ADP
ap-7678	503	26	e	e	PROPN
ap-7678	503	27	=	=	SYM
ap-7678	503	28	49.51148774312169	49.51148774312169	NUM
ap-7678	503	29	(	(	PUNCT
ap-7678	503	30	i.e.	i.e.	X
ap-7678	503	31	n	n	CCONJ
ap-7678	503	32	=	=	SYM
ap-7678	503	33	50	50	NUM
ap-7678	503	34	)	)	PUNCT
ap-7678	503	35	,	,	PUNCT
ap-7678	503	36	lies	lie	VERB
ap-7678	503	37	outside	outside	ADV
ap-7678	503	38	of	of	ADP
ap-7678	503	39	the	the	DET
ap-7678	503	40	bounds	bound	NOUN
ap-7678	503	41	generated	generate	VERB
ap-7678	503	42	in	in	ADP
ap-7678	503	43	the	the	DET
ap-7678	503	44	last	last	ADJ
ap-7678	503	45	two	two	NUM
ap-7678	503	46	columns	column	NOUN
ap-7678	503	47	,	,	PUNCT
ap-7678	503	48	for	for	ADP
ap-7678	503	49	n	n	X
ap-7678	503	50	>	>	X
ap-7678	503	51	80	80	NUM
ap-7678	503	52	.	.	PUNCT
ap-7678	504	1	however	however	ADV
ap-7678	504	2	,	,	PUNCT
ap-7678	504	3	all	all	DET
ap-7678	504	4	oppq	oppq	NOUN
ap-7678	504	5	-	-	PUNCT
ap-7678	504	6	bm	bm	PROPN
ap-7678	504	7	estimates	estimate	NOUN
ap-7678	504	8	must	must	AUX
ap-7678	504	9	lie	lie	VERB
ap-7678	504	10	within	within	ADP
ap-7678	504	11	the	the	DET
ap-7678	504	12	bounds	bound	NOUN
ap-7678	504	13	calculated	calculate	VERB
ap-7678	504	14	at	at	ADP
ap-7678	504	15	lower	low	ADJ
ap-7678	504	16	order	order	NOUN
ap-7678	504	17	.	.	PUNCT
ap-7678	505	1	9.6	9.6	NUM
ap-7678	505	2	.	.	PUNCT
ap-7678	506	1	comparison	comparison	NOUN
ap-7678	506	2	with	with	ADP
ap-7678	506	3	emm	emm	PROPN
ap-7678	506	4	bounds	bound	NOUN
ap-7678	506	5	generally	generally	ADV
ap-7678	506	6	,	,	PUNCT
ap-7678	506	7	the	the	DET
ap-7678	506	8	emm	emm	PROPN
ap-7678	506	9	analysis	analysis	NOUN
ap-7678	506	10	will	will	AUX
ap-7678	506	11	be	be	AUX
ap-7678	506	12	more	more	ADV
ap-7678	506	13	efficient	efficient	ADJ
ap-7678	506	14	in	in	ADP
ap-7678	506	15	generating	generate	VERB
ap-7678	506	16	bounds	bound	NOUN
ap-7678	506	17	than	than	ADP
ap-7678	506	18	oppq	oppq	NOUN
ap-7678	506	19	-	-	PUNCT
ap-7678	506	20	bm	bm	PROPN
ap-7678	506	21	.	.	PUNCT
ap-7678	507	1	that	that	PRON
ap-7678	507	2	is	be	AUX
ap-7678	507	3	,	,	PUNCT
ap-7678	507	4	fewer	few	ADJ
ap-7678	507	5	moments	moment	NOUN
ap-7678	507	6	will	will	AUX
ap-7678	507	7	be	be	AUX
ap-7678	507	8	required	require	VERB
ap-7678	507	9	to	to	PART
ap-7678	507	10	generate	generate	VERB
ap-7678	507	11	the	the	DET
ap-7678	507	12	same	same	ADJ
ap-7678	507	13	level	level	NOUN
ap-7678	507	14	of	of	ADP
ap-7678	507	15	tightness	tightness	NOUN
ap-7678	507	16	of	of	ADP
ap-7678	507	17	the	the	DET
ap-7678	507	18	bounds	bound	NOUN
ap-7678	507	19	.	.	PUNCT
ap-7678	508	1	however	however	ADV
ap-7678	508	2	,	,	PUNCT
ap-7678	508	3	this	this	PRON
ap-7678	508	4	depends	depend	VERB
ap-7678	508	5	74	74	NUM
ap-7678	508	6	vol	vol	NOUN
ap-7678	508	7	.	.	PUNCT
ap-7678	509	1	62	62	NUM
ap-7678	509	2	no	no	INTJ
ap-7678	509	3	.	.	PUNCT
ap-7678	510	1	1/2022	1/2022	NUM
ap-7678	510	2	algebraic	algebraic	ADJ
ap-7678	510	3	eigenenergy	eigenenergy	NOUN
ap-7678	510	4	bounding	bounding	NOUN
ap-7678	510	5	method	method	NOUN
ap-7678	510	6	n	n	ADP
ap-7678	510	7	e	e	X
ap-7678	510	8	(	(	PUNCT
ap-7678	510	9	l	l	NOUN
ap-7678	510	10	)	)	PUNCT
ap-7678	510	11	8	8	NUM
ap-7678	510	12	e	e	NOUN
ap-7678	510	13	(	(	PUNCT
ap-7678	510	14	u	u	NOUN
ap-7678	510	15	)	)	PUNCT
ap-7678	510	16	8	8	NUM
ap-7678	510	17	150	150	NUM
ap-7678	510	18	49.511500377377302482	49.511500377377302482	NUM
ap-7678	510	19	49.511500377382545933	49.511500377382545933	NUM
ap-7678	510	20	160	160	NUM
ap-7678	510	21	49.511500377379459378	49.511500377379459378	NUM
ap-7678	510	22	49.511500377380389040	49.511500377380389040	NUM
ap-7678	510	23	170	170	NUM
ap-7678	510	24	49.511500377379839286	49.511500377379839286	NUM
ap-7678	510	25	49.511500377380009132	49.511500377380009132	NUM
ap-7678	510	26	180	180	NUM
ap-7678	510	27	49.511500377379908253	49.511500377379908253	NUM
ap-7678	510	28	49.511500377379940166	49.511500377379940166	NUM
ap-7678	510	29	190	190	NUM
ap-7678	510	30	49.511500377379921131	49.511500377379921131	NUM
ap-7678	510	31	49.511500377379927287	49.511500377379927287	NUM
ap-7678	510	32	200	200	NUM
ap-7678	510	33	49.511500377379923601	49.511500377379923601	NUM
ap-7678	510	34	49.511500377379924818	49.511500377379924818	NUM
ap-7678	510	35	210	210	NUM
ap-7678	510	36	49.511500377379924086	49.511500377379924086	NUM
ap-7678	510	37	49.511500377379924333	49.511500377379924333	NUM
ap-7678	510	38	220	220	NUM
ap-7678	510	39	49.511500377379924184	49.511500377379924184	NUM
ap-7678	510	40	49.511500377379924235	49.511500377379924235	NUM
ap-7678	510	41	230	230	NUM
ap-7678	510	42	49.511500377379924204	49.511500377379924204	NUM
ap-7678	510	43	49.511500377379924215	49.511500377379924215	NUM
ap-7678	510	44	240	240	NUM
ap-7678	510	45	49.511500377379924208	49.511500377379924208	NUM
ap-7678	510	46	49.511500377379924210	49.511500377379924210	NUM
ap-7678	510	47	250	250	NUM
ap-7678	510	48	49.511500377379924209048830	49.511500377379924209048830	NUM
ap-7678	510	49	49.511500377379924209558953	49.511500377379924209558953	NUM
ap-7678	510	50	260	260	NUM
ap-7678	510	51	49.511500377379924209246851	49.511500377379924209246851	NUM
ap-7678	510	52	49.511500377379924209360932	49.511500377379924209360932	NUM
ap-7678	510	53	270	270	NUM
ap-7678	510	54	49.511500377379924209290914	49.511500377379924209290914	NUM
ap-7678	510	55	49.511500377379924209316869	49.511500377379924209316869	NUM
ap-7678	510	56	280	280	NUM
ap-7678	510	57	49.511500377379924209300890	49.511500377379924209300890	NUM
ap-7678	510	58	49.511500377379924209306893	49.511500377379924209306893	NUM
ap-7678	510	59	290	290	NUM
ap-7678	510	60	49.511500377379924209303186	49.511500377379924209303186	NUM
ap-7678	510	61	49.511500377379924209304597	49.511500377379924209304597	NUM
ap-7678	510	62	300	300	NUM
ap-7678	510	63	49.511500377379924209303723	49.511500377379924209303723	NUM
ap-7678	510	64	49.511500377379924209304060	49.511500377379924209304060	NUM
ap-7678	510	65	b	b	X
ap-7678	510	66	=	=	SYM
ap-7678	510	67	8.01	8.01	NUM
ap-7678	510	68	table	table	NOUN
ap-7678	510	69	7	7	NUM
ap-7678	510	70	.	.	PUNCT
ap-7678	510	71	bounds	bound	NOUN
ap-7678	510	72	for	for	ADP
ap-7678	510	73	e8	e8	PROPN
ap-7678	510	74	based	base	VERB
ap-7678	510	75	on	on	ADP
ap-7678	510	76	chosen	choose	VERB
ap-7678	510	77	b(u	b(u	PROPN
ap-7678	510	78	)	)	PUNCT
ap-7678	510	79	.	.	PUNCT
ap-7678	511	1	n	n	X
ap-7678	511	2	e0	e0	PROPN
ap-7678	511	3	e2	e2	PROPN
ap-7678	511	4	e4	e4	PROPN
ap-7678	511	5	e6	e6	PROPN
ap-7678	511	6	e8	e8	PROPN
ap-7678	511	7	1	1	NUM
ap-7678	511	8	-0.5232686221275529	-0.5232686221275529	NOUN
ap-7678	511	9	5	5	NUM
ap-7678	511	10	-0.5232686221276021	-0.5232686221276021	NOUN
ap-7678	511	11	5.37847969429	5.37847969429	NUM
ap-7678	511	12	16.78604192264	16.78604192264	NUM
ap-7678	511	13	10	10	NUM
ap-7678	511	14	-0.5232686221275522	-0.5232686221275522	VERB
ap-7678	511	15	5.37497046837	5.37497046837	NUM
ap-7678	511	16	16.79542276464	16.79542276464	NUM
ap-7678	511	17	31.74535880954	31.74535880954	NUM
ap-7678	511	18	49.5248113799	49.5248113799	NUM
ap-7678	511	19	20	20	NUM
ap-7678	511	20	-0.5232686221275523	-0.5232686221275523	NOUN
ap-7678	511	21	5.37497000884	5.37497000884	NUM
ap-7678	511	22	16.79534683270	16.79534683270	NUM
ap-7678	511	23	31.74254983720	31.74254983720	NUM
ap-7678	511	24	49.5115003841	49.5115003841	NUM
ap-7678	511	25	30	30	NUM
ap-7678	511	26	-0.5232686221275523	-0.5232686221275523	NOUN
ap-7678	511	27	5.37497000884	5.37497000884	NUM
ap-7678	511	28	16.79534683270	16.79534683270	NUM
ap-7678	511	29	31.74254983711	31.74254983711	NUM
ap-7678	511	30	49.5115003772	49.5115003772	NUM
ap-7678	511	31	table	table	NOUN
ap-7678	511	32	8	8	NUM
ap-7678	511	33	.	.	PUNCT
ap-7678	512	1	oppq	oppq	PROPN
ap-7678	512	2	-	-	PUNCT
ap-7678	512	3	bm	bm	PROPN
ap-7678	512	4	estimates	estimate	NOUN
ap-7678	512	5	(	(	PUNCT
ap-7678	512	6	i.e.	i.e.	X
ap-7678	512	7	∂esn	∂esn	X
ap-7678	512	8	(	(	PUNCT
ap-7678	512	9	e	e	NOUN
ap-7678	512	10	)	)	PUNCT
ap-7678	512	11	=	=	SYM
ap-7678	512	12	0	0	NUM
ap-7678	512	13	)	)	PUNCT
ap-7678	512	14	for	for	ADP
ap-7678	512	15	v	v	NOUN
ap-7678	512	16	(	(	PUNCT
ap-7678	512	17	x	x	NOUN
ap-7678	512	18	)	)	PUNCT
ap-7678	512	19	=	=	SYM
ap-7678	512	20	x6	x6	PROPN
ap-7678	512	21	−	−	PROPN
ap-7678	512	22	4x2	4x2	NUM
ap-7678	512	23	,	,	PUNCT
ap-7678	512	24	ms	ms	PROPN
ap-7678	512	25	=	=	SYM
ap-7678	512	26	0	0	PROPN
ap-7678	512	27	,	,	PUNCT
ap-7678	512	28	r(x	r(x	NOUN
ap-7678	512	29	)	)	PUNCT
ap-7678	512	30	=	=	SYM
ap-7678	512	31	φgr(x	φgr(x	PROPN
ap-7678	512	32	)	)	PUNCT
ap-7678	512	33	based	base	VERB
ap-7678	512	34	on	on	ADP
ap-7678	512	35	the	the	DET
ap-7678	512	36	first	first	ADJ
ap-7678	512	37	61	61	NUM
ap-7678	512	38	,	,	PUNCT
ap-7678	512	39	emm	emm	PROPN
ap-7678	512	40	generated	generate	VERB
ap-7678	512	41	,	,	PUNCT
ap-7678	512	42	ground	ground	NOUN
ap-7678	512	43	state	state	NOUN
ap-7678	512	44	power	power	NOUN
ap-7678	512	45	moments	moment	NOUN
ap-7678	512	46	{	{	PUNCT
ap-7678	512	47	ugr(p	ugr(p	PROPN
ap-7678	512	48	≤	≤	NUM
ap-7678	512	49	60	60	NUM
ap-7678	512	50	)	)	PUNCT
ap-7678	512	51	}	}	PUNCT
ap-7678	512	52	.	.	PUNCT
ap-7678	513	1	on	on	ADP
ap-7678	513	2	the	the	DET
ap-7678	513	3	choice	choice	NOUN
ap-7678	513	4	of	of	ADP
ap-7678	513	5	weight	weight	NOUN
ap-7678	513	6	,	,	PUNCT
ap-7678	513	7	as	as	SCONJ
ap-7678	513	8	the	the	DET
ap-7678	513	9	following	follow	VERB
ap-7678	513	10	case	case	NOUN
ap-7678	513	11	exemplifies	exemplify	VERB
ap-7678	513	12	.	.	PUNCT
ap-7678	514	1	thus	thus	ADV
ap-7678	514	2	,	,	PUNCT
ap-7678	514	3	with	with	ADP
ap-7678	514	4	regards	regard	NOUN
ap-7678	514	5	to	to	ADP
ap-7678	514	6	the	the	DET
ap-7678	514	7	ms	ms	NOUN
ap-7678	514	8	=	=	PROPN
ap-7678	514	9	2	2	NUM
ap-7678	514	10	missing	miss	VERB
ap-7678	514	11	moment	moment	NOUN
ap-7678	514	12	formulation	formulation	NOUN
ap-7678	514	13	given	give	VERB
ap-7678	514	14	earlier	early	ADV
ap-7678	514	15	,	,	PUNCT
ap-7678	514	16	the	the	DET
ap-7678	514	17	emm	emm	PROPN
ap-7678	514	18	bounds	bound	VERB
ap-7678	514	19	for	for	ADP
ap-7678	514	20	the	the	DET
ap-7678	514	21	ground	ground	NOUN
ap-7678	514	22	state	state	NOUN
ap-7678	515	1	−0.523268623844284	−0.523268623844284	PROPN
ap-7678	515	2	<	<	X
ap-7678	515	3	egr	egr	PROPN
ap-7678	515	4	<	<	X
ap-7678	515	5	−0.523268619253327	−0.523268619253327	PROPN
ap-7678	515	6	,	,	PUNCT
ap-7678	515	7	were	be	AUX
ap-7678	515	8	based	base	VERB
ap-7678	515	9	on	on	ADP
ap-7678	515	10	the	the	DET
ap-7678	515	11	first	first	ADJ
ap-7678	515	12	29	29	NUM
ap-7678	515	13	power	power	NOUN
ap-7678	515	14	moments	moment	NOUN
ap-7678	515	15	.	.	PUNCT
ap-7678	516	1	the	the	DET
ap-7678	516	2	same	same	ADJ
ap-7678	516	3	(	(	PUNCT
ap-7678	516	4	approximately	approximately	ADV
ap-7678	516	5	)	)	PUNCT
ap-7678	516	6	level	level	NOUN
ap-7678	516	7	of	of	ADP
ap-7678	516	8	tightness	tightness	NOUN
ap-7678	516	9	was	be	AUX
ap-7678	516	10	achieved	achieve	VERB
ap-7678	516	11	with	with	ADP
ap-7678	516	12	oppq	oppq	NOUN
ap-7678	516	13	-	-	PUNCT
ap-7678	516	14	bm	bm	PROPN
ap-7678	516	15	using	use	VERB
ap-7678	516	16	more	more	ADJ
ap-7678	516	17	than	than	ADP
ap-7678	516	18	50	50	NUM
ap-7678	516	19	power	power	NOUN
ap-7678	516	20	moments	moment	NOUN
ap-7678	516	21	,	,	PUNCT
ap-7678	516	22	based	base	VERB
ap-7678	516	23	on	on	ADP
ap-7678	516	24	the	the	DET
ap-7678	516	25	results	result	NOUN
ap-7678	516	26	in	in	ADP
ap-7678	516	27	table	table	NOUN
ap-7678	516	28	3	3	NUM
ap-7678	516	29	.	.	PUNCT
ap-7678	517	1	thus	thus	ADV
ap-7678	517	2	,	,	PUNCT
ap-7678	517	3	in	in	ADP
ap-7678	517	4	this	this	DET
ap-7678	517	5	example	example	NOUN
ap-7678	517	6	emm	emm	PROPN
ap-7678	517	7	is	be	AUX
ap-7678	517	8	vastly	vastly	ADV
ap-7678	517	9	superior	superior	ADJ
ap-7678	517	10	to	to	ADP
ap-7678	517	11	oppq	oppq	NOUN
ap-7678	517	12	-	-	PUNCT
ap-7678	517	13	bm	bm	PROPN
ap-7678	517	14	.	.	PUNCT
ap-7678	518	1	however	however	ADV
ap-7678	518	2	,	,	PUNCT
ap-7678	518	3	for	for	ADP
ap-7678	518	4	the	the	DET
ap-7678	518	5	next	next	ADJ
ap-7678	518	6	example	example	NOUN
ap-7678	518	7	the	the	DET
ap-7678	518	8	situation	situation	NOUN
ap-7678	518	9	significantly	significantly	ADV
ap-7678	518	10	improves	improve	VERB
ap-7678	518	11	for	for	ADP
ap-7678	518	12	oppq	oppq	NOUN
ap-7678	518	13	-	-	PUNCT
ap-7678	518	14	bm	bm	PROPN
ap-7678	518	15	over	over	ADP
ap-7678	518	16	emm	emm	PROPN
ap-7678	518	17	.	.	PUNCT
ap-7678	519	1	for	for	ADP
ap-7678	519	2	the	the	DET
ap-7678	519	3	ms	ms	PROPN
ap-7678	519	4	=	=	SYM
ap-7678	519	5	0	0	NUM
ap-7678	519	6	formulation	formulation	NOUN
ap-7678	519	7	being	be	AUX
ap-7678	519	8	considered	consider	VERB
ap-7678	519	9	,	,	PUNCT
ap-7678	519	10	using	use	VERB
ap-7678	519	11	the	the	DET
ap-7678	519	12	first	first	ADJ
ap-7678	519	13	62	62	NUM
ap-7678	519	14	stieljes	stieljes	NOUN
ap-7678	519	15	power	power	NOUN
ap-7678	519	16	moments	moment	NOUN
ap-7678	519	17	(	(	PUNCT
ap-7678	519	18	i.e.	i.e.	X
ap-7678	519	19	{	{	PUNCT
ap-7678	519	20	u(p)|p	u(p)|p	NOUN
ap-7678	519	21	≤	≤	NOUN
ap-7678	519	22	61	61	NUM
ap-7678	519	23	}	}	PUNCT
ap-7678	519	24	)	)	PUNCT
ap-7678	519	25	,	,	PUNCT
ap-7678	519	26	emm	emm	PROPN
ap-7678	519	27	achieves	achieve	VERB
ap-7678	519	28	the	the	DET
ap-7678	519	29	bounds	bound	NOUN
ap-7678	519	30	:	:	PUNCT
ap-7678	519	31	−	−	PROPN
ap-7678	519	32	.52326862212755223941616949719078449	.52326862212755223941616949719078449	NOUN
ap-7678	519	33	<	<	X
ap-7678	519	34	eemm	eemm	NOUN
ap-7678	519	35	<	<	X
ap-7678	519	36	−.52326862212755223941616949719078395	−.52326862212755223941616949719078395	PROPN
ap-7678	519	37	(	(	PUNCT
ap-7678	519	38	69	69	NUM
ap-7678	519	39	)	)	PUNCT
ap-7678	519	40	using	use	VERB
ap-7678	519	41	the	the	DET
ap-7678	519	42	weight	weight	NOUN
ap-7678	519	43	rl	rl	ADP
ap-7678	519	44	the	the	DET
ap-7678	519	45	oppq	oppq	NOUN
ap-7678	519	46	-	-	PUNCT
ap-7678	519	47	am	am	NOUN
ap-7678	519	48	estimate	estimate	NOUN
ap-7678	519	49	achieves	achieve	VERB
ap-7678	519	50	this	this	DET
ap-7678	519	51	level	level	NOUN
ap-7678	519	52	of	of	ADP
ap-7678	519	53	accuracy	accuracy	NOUN
ap-7678	519	54	on	on	ADP
ap-7678	519	55	the	the	DET
ap-7678	519	56	basis	basis	NOUN
ap-7678	519	57	of	of	ADP
ap-7678	519	58	approximately	approximately	ADV
ap-7678	519	59	220	220	NUM
ap-7678	519	60	power	power	NOUN
ap-7678	519	61	moments	moment	NOUN
ap-7678	519	62	;	;	PUNCT
ap-7678	519	63	whereas	whereas	SCONJ
ap-7678	519	64	the	the	DET
ap-7678	519	65	weight	weight	NOUN
ap-7678	519	66	rg	rg	PROPN
ap-7678	519	67	generates	generate	VERB
ap-7678	519	68	an	an	DET
ap-7678	519	69	oppq	oppq	NOUN
ap-7678	519	70	-	-	PUNCT
ap-7678	519	71	am	am	NOUN
ap-7678	519	72	estimate	estimate	NOUN
ap-7678	519	73	that	that	PRON
ap-7678	519	74	surpasses	surpass	VERB
ap-7678	519	75	the	the	DET
ap-7678	519	76	emm	emm	PROPN
ap-7678	519	77	accuracy	accuracy	NOUN
ap-7678	519	78	based	base	VERB
ap-7678	519	79	only	only	ADV
ap-7678	519	80	on	on	ADP
ap-7678	519	81	the	the	DET
ap-7678	519	82	use	use	NOUN
ap-7678	519	83	of	of	ADP
ap-7678	519	84	45	45	NUM
ap-7678	519	85	moments	moment	NOUN
ap-7678	519	86	(	(	PUNCT
ap-7678	519	87	oppqam	oppqam	PROPN
ap-7678	519	88	):	):	PUNCT
ap-7678	519	89	eop	eop	PROPN
ap-7678	519	90	p	p	PROPN
ap-7678	519	91	q−am	q−am	PROPN
ap-7678	519	92	=	=	SYM
ap-7678	519	93	−0.523268622127552239416169497	−0.523268622127552239416169497	PROPN
ap-7678	519	94	19078406116564771630604	19078406116564771630604	NUM
ap-7678	519	95	(	(	PUNCT
ap-7678	519	96	70	70	NUM
ap-7678	519	97	)	)	PUNCT
ap-7678	519	98	9.7	9.7	NUM
ap-7678	519	99	.	.	PUNCT
ap-7678	520	1	using	use	VERB
ap-7678	520	2	the	the	DET
ap-7678	520	3	ground	ground	NOUN
ap-7678	520	4	state	state	NOUN
ap-7678	520	5	as	as	ADP
ap-7678	520	6	a	a	DET
ap-7678	520	7	weight	weight	NOUN
ap-7678	520	8	in	in	ADP
ap-7678	520	9	table	table	NOUN
ap-7678	520	10	8	8	NUM
ap-7678	520	11	we	we	PRON
ap-7678	520	12	implement	implement	VERB
ap-7678	520	13	oppq	oppq	NOUN
ap-7678	520	14	-	-	PUNCT
ap-7678	520	15	bm	bm	PROPN
ap-7678	520	16	on	on	ADP
ap-7678	520	17	a	a	DET
ap-7678	520	18	representation	representation	NOUN
ap-7678	520	19	that	that	PRON
ap-7678	520	20	uses	use	VERB
ap-7678	520	21	the	the	DET
ap-7678	520	22	generated	generate	VERB
ap-7678	520	23	moments	moment	NOUN
ap-7678	520	24	of	of	ADP
ap-7678	520	25	the	the	DET
ap-7678	520	26	unknown	unknown	ADJ
ap-7678	520	27	ground	ground	NOUN
ap-7678	520	28	state	state	PROPN
ap-7678	520	29	r(x	r(x	PROPN
ap-7678	520	30	)	)	PUNCT
ap-7678	520	31	=	=	SYM
ap-7678	520	32	ψgr(x	ψgr(x	PROPN
ap-7678	520	33	)	)	PUNCT
ap-7678	520	34	.	.	PUNCT
ap-7678	521	1	as	as	SCONJ
ap-7678	521	2	expected	expect	VERB
ap-7678	521	3	,	,	PUNCT
ap-7678	521	4	the	the	DET
ap-7678	521	5	convergence	convergence	NOUN
ap-7678	521	6	is	be	AUX
ap-7678	521	7	very	very	ADV
ap-7678	521	8	fast	fast	ADJ
ap-7678	521	9	in	in	ADP
ap-7678	521	10	comparison	comparison	NOUN
ap-7678	521	11	with	with	ADP
ap-7678	521	12	results	result	NOUN
ap-7678	521	13	based	base	VERB
ap-7678	521	14	on	on	ADP
ap-7678	521	15	rl	rl	PROPN
ap-7678	521	16	.	.	PROPN
ap-7678	521	17	9.8	9.8	NUM
ap-7678	521	18	.	.	PUNCT
ap-7678	522	1	emm	emm	PROPN
ap-7678	522	2	results	result	VERB
ap-7678	522	3	for	for	ADP
ap-7678	522	4	the	the	DET
ap-7678	522	5	probability	probability	NOUN
ap-7678	522	6	density	density	NOUN
ap-7678	522	7	application	application	NOUN
ap-7678	522	8	of	of	ADP
ap-7678	522	9	emm	emm	PROPN
ap-7678	522	10	to	to	ADP
ap-7678	522	11	the	the	DET
ap-7678	522	12	probability	probability	NOUN
ap-7678	522	13	density	density	NOUN
ap-7678	522	14	(	(	PUNCT
ap-7678	522	15	i.e.	i.e.	X
ap-7678	522	16	emm	emm	PROPN
ap-7678	522	17	-	-	PUNCT
ap-7678	522	18	ψ2	ψ2	NOUN
ap-7678	522	19	,	,	PUNCT
ap-7678	522	20	an	an	DET
ap-7678	522	21	ms	ms	NOUN
ap-7678	522	22	=	=	SYM
ap-7678	522	23	2	2	NUM
ap-7678	522	24	problem	problem	NOUN
ap-7678	522	25	)	)	PUNCT
ap-7678	522	26	yields	yield	VERB
ap-7678	522	27	the	the	DET
ap-7678	522	28	bounds	bound	NOUN
ap-7678	522	29	−.52326866	−.52326866	PROPN
ap-7678	522	30	<	<	X
ap-7678	522	31	e0	e0	PROPN
ap-7678	522	32	<	<	X
ap-7678	522	33	−.52326857	−.52326857	PROPN
ap-7678	522	34	,	,	PUNCT
ap-7678	522	35	1.0057681	1.0057681	NUM
ap-7678	522	36	<	<	X
ap-7678	522	37	e1	e1	PROPN
ap-7678	522	38	<	<	X
ap-7678	522	39	1.0057685	1.0057685	NUM
ap-7678	522	40	,	,	PUNCT
ap-7678	522	41	5.3749699	5.3749699	NUM
ap-7678	522	42	<	<	X
ap-7678	522	43	e2	e2	PROPN
ap-7678	522	44	<	<	X
ap-7678	522	45	5.3749701	5.3749701	NUM
ap-7678	522	46	,	,	PUNCT
ap-7678	522	47	10.5725845	10.5725845	NUM
ap-7678	522	48	<	<	X
ap-7678	522	49	e3	e3	X
ap-7678	522	50	<	<	X
ap-7678	522	51	10.5725855	10.5725855	NUM
ap-7678	522	52	,	,	PUNCT
ap-7678	522	53	16.795339	16.795339	NUM
ap-7678	522	54	<	<	X
ap-7678	522	55	e4	e4	PROPN
ap-7678	522	56	<	<	X
ap-7678	522	57	16.795351	16.795351	NUM
ap-7678	522	58	,	,	PUNCT
ap-7678	522	59	23.883886	23.883886	NUM
ap-7678	522	60	<	<	X
ap-7678	522	61	e5	e5	PROPN
ap-7678	522	62	<	<	X
ap-7678	522	63	23.883961	23.883961	NUM
ap-7678	522	64	,	,	PUNCT
ap-7678	522	65	31.74217	31.74217	NUM
ap-7678	522	66	<	<	X
ap-7678	522	67	e6	e6	PROPN
ap-7678	522	68	<	<	X
ap-7678	522	69	31.74323	31.74323	NUM
ap-7678	522	70	,	,	PUNCT
ap-7678	522	71	40.301	40.301	NUM
ap-7678	522	72	<	<	X
ap-7678	522	73	e7	e7	PROPN
ap-7678	522	74	<	<	X
ap-7678	522	75	40.305	40.305	NUM
ap-7678	522	76	,	,	PUNCT
ap-7678	522	77	49.506	49.506	NUM
ap-7678	522	78	<	<	X
ap-7678	522	79	e8	e8	PROPN
ap-7678	522	80	<	<	X
ap-7678	522	81	49.533	49.533	NUM
ap-7678	522	82	all	all	PRON
ap-7678	522	83	to	to	ADP
ap-7678	522	84	the	the	DET
ap-7678	522	85	same	same	ADJ
ap-7678	522	86	moment	moment	NOUN
ap-7678	522	87	expansion	expansion	NOUN
ap-7678	522	88	order	order	NOUN
ap-7678	522	89	of	of	ADP
ap-7678	522	90	28	28	NUM
ap-7678	522	91	.	.	PUNCT
ap-7678	523	1	we	we	PRON
ap-7678	523	2	also	also	ADV
ap-7678	523	3	note	note	VERB
ap-7678	523	4	that	that	SCONJ
ap-7678	523	5	an	an	PRON
ap-7678	523	6	emm	emm	PROPN
ap-7678	523	7	-	-	PUNCT
ap-7678	523	8	ψ{ms	ψ{ms	PROPN
ap-7678	523	9	=	=	SYM
ap-7678	523	10	2	2	X
ap-7678	523	11	}	}	PUNCT
ap-7678	523	12	bounding	bound	VERB
ap-7678	523	13	formulation	formulation	NOUN
ap-7678	523	14	on	on	ADP
ap-7678	523	15	the	the	DET
ap-7678	523	16	ground	ground	NOUN
ap-7678	523	17	state	state	NOUN
ap-7678	523	18	yields	yield	NOUN
ap-7678	523	19	comparable	comparable	ADJ
ap-7678	523	20	bounds	bound	NOUN
ap-7678	523	21	using	use	VERB
ap-7678	523	22	the	the	DET
ap-7678	523	23	first	first	ADJ
ap-7678	523	24	28	28	NUM
ap-7678	523	25	power	power	NOUN
ap-7678	523	26	moments	moment	NOUN
ap-7678	523	27	:	:	PUNCT
ap-7678	523	28	−.5232686237	−.5232686237	PROPN
ap-7678	523	29	<	<	X
ap-7678	523	30	egr	egr	X
ap-7678	523	31	<	<	X
ap-7678	523	32	−.5232686193	−.5232686193	PROPN
ap-7678	523	33	.	.	PUNCT
ap-7678	524	1	with	with	ADP
ap-7678	524	2	regards	regard	NOUN
ap-7678	524	3	75	75	NUM
ap-7678	524	4	carlos	carlos	PROPN
ap-7678	524	5	r.	r.	PROPN
ap-7678	524	6	handy	handy	PROPN
ap-7678	524	7	acta	acta	PROPN
ap-7678	524	8	polytechnica	polytechnica	PROPN
ap-7678	524	9	to	to	ADP
ap-7678	524	10	e8	e8	PROPN
ap-7678	524	11	,	,	PUNCT
ap-7678	524	12	extension	extension	NOUN
ap-7678	524	13	of	of	ADP
ap-7678	524	14	emm	emm	PROPN
ap-7678	524	15	-	-	PUNCT
ap-7678	524	16	ψ2{ms	ψ2{ms	NUM
ap-7678	524	17	=	=	SYM
ap-7678	524	18	2	2	X
ap-7678	524	19	}	}	PUNCT
ap-7678	524	20	analysis	analysis	NOUN
ap-7678	524	21	to	to	ADP
ap-7678	524	22	the	the	DET
ap-7678	524	23	first	first	ADJ
ap-7678	524	24	48	48	NUM
ap-7678	524	25	moments	moment	NOUN
ap-7678	524	26	(	(	PUNCT
ap-7678	524	27	i.e.	i.e.	X
ap-7678	524	28	{	{	PUNCT
ap-7678	524	29	u(p)|0	u(p)|0	ADJ
ap-7678	524	30	≤	≤	PROPN
ap-7678	524	31	p	p	NOUN
ap-7678	524	32	≤	≤	NUM
ap-7678	524	33	47	47	NUM
ap-7678	524	34	}	}	PUNCT
ap-7678	524	35	)	)	PUNCT
ap-7678	524	36	yields	yield	VERB
ap-7678	524	37	the	the	DET
ap-7678	524	38	bounds	bound	NOUN
ap-7678	524	39	49.5115003768	49.5115003768	NUM
ap-7678	524	40	<	<	X
ap-7678	524	41	e8	e8	PROPN
ap-7678	524	42	<	<	X
ap-7678	524	43	49.5115003798	49.5115003798	NUM
ap-7678	524	44	.	.	PUNCT
ap-7678	525	1	these	these	PRON
ap-7678	525	2	serve	serve	VERB
ap-7678	525	3	to	to	PART
ap-7678	525	4	further	far	ADV
ap-7678	525	5	confirm	confirm	VERB
ap-7678	525	6	some	some	PRON
ap-7678	525	7	of	of	ADP
ap-7678	525	8	the	the	DET
ap-7678	525	9	results	result	NOUN
ap-7678	525	10	previously	previously	ADV
ap-7678	525	11	cited	cite	VERB
ap-7678	525	12	.	.	PUNCT
ap-7678	526	1	10	10	NUM
ap-7678	526	2	.	.	PUNCT
ap-7678	527	1	a	a	DET
ap-7678	527	2	pt	pt	PROPN
ap-7678	527	3	-	-	PUNCT
ap-7678	527	4	symmetry	symmetry	NOUN
ap-7678	527	5	breaking	breaking	NOUN
ap-7678	527	6	,	,	PUNCT
ap-7678	527	7	non	non	ADJ
ap-7678	527	8	-	-	ADJ
ap-7678	527	9	hermitian	hermitian	ADJ
ap-7678	527	10	,	,	PUNCT
ap-7678	527	11	problem	problem	NOUN
ap-7678	527	12	consider	consider	VERB
ap-7678	527	13	the	the	DET
ap-7678	527	14	non	non	ADJ
ap-7678	527	15	-	-	ADJ
ap-7678	527	16	hermitian	hermitian	ADJ
ap-7678	527	17	system	system	NOUN
ap-7678	527	18	−∂2	−∂2	PROPN
ap-7678	527	19	xψ(x	xψ(x	NUM
ap-7678	527	20	)	)	PUNCT
ap-7678	528	1	+	+	CCONJ
ap-7678	528	2	(	(	PUNCT
ap-7678	528	3	ix3	ix3	PROPN
ap-7678	528	4	+	+	CCONJ
ap-7678	528	5	iax)ψ(x	iax)ψ(x	PROPN
ap-7678	528	6	)	)	PUNCT
ap-7678	528	7	=	=	SYM
ap-7678	528	8	eψ(x	eψ(x	NOUN
ap-7678	528	9	)	)	PUNCT
ap-7678	528	10	,	,	PUNCT
ap-7678	528	11	(	(	PUNCT
ap-7678	528	12	71	71	NUM
ap-7678	528	13	)	)	PUNCT
ap-7678	528	14	which	which	PRON
ap-7678	528	15	is	be	AUX
ap-7678	528	16	known	know	VERB
ap-7678	528	17	to	to	PART
ap-7678	528	18	break	break	VERB
ap-7678	528	19	pt	pt	NOUN
ap-7678	528	20	symmetry	symmetry	NOUN
ap-7678	528	21	for	for	ADP
ap-7678	528	22	negative	negative	ADJ
ap-7678	528	23	values	value	NOUN
ap-7678	528	24	of	of	ADP
ap-7678	528	25	the	the	DET
ap-7678	528	26	‘	'	PUNCT
ap-7678	528	27	a	a	DET
ap-7678	528	28	’	'	PUNCT
ap-7678	528	29	parameter	parameter	NOUN
ap-7678	528	30	(	(	PUNCT
ap-7678	528	31	a	a	DET
ap-7678	528	32	<	<	X
ap-7678	528	33	−2.611809356	−2.611809356	NOUN
ap-7678	528	34	,	,	PUNCT
ap-7678	528	35	and	and	CCONJ
ap-7678	528	36	for	for	ADP
ap-7678	528	37	a	a	DET
ap-7678	528	38	<	<	X
ap-7678	528	39	−5.375879629	−5.375879629	NOUN
ap-7678	528	40	)	)	PUNCT
ap-7678	528	41	from	from	ADP
ap-7678	528	42	refs	ref	NOUN
ap-7678	528	43	.	.	PUNCT
ap-7678	529	1	[	[	X
ap-7678	529	2	8	8	NUM
ap-7678	529	3	,	,	PUNCT
ap-7678	529	4	25	25	NUM
ap-7678	529	5	]	]	PUNCT
ap-7678	529	6	.	.	PUNCT
ap-7678	530	1	the	the	DET
ap-7678	530	2	case	case	NOUN
ap-7678	530	3	a	a	DET
ap-7678	530	4	=	=	SYM
ap-7678	530	5	0	0	PUNCT
ap-7678	530	6	has	have	VERB
ap-7678	530	7	purely	purely	ADV
ap-7678	530	8	real	real	ADJ
ap-7678	530	9	eigen	eigen	NOUN
ap-7678	530	10	-	-	PUNCT
ap-7678	530	11	energies	energy	NOUN
ap-7678	530	12	,	,	PUNCT
ap-7678	530	13	as	as	SCONJ
ap-7678	530	14	computed	compute	VERB
ap-7678	530	15	by	by	ADP
ap-7678	530	16	bender	bender	NOUN
ap-7678	530	17	and	and	CCONJ
ap-7678	530	18	boettcher	boettcher	PRON
ap-7678	531	1	[	[	X
ap-7678	531	2	5	5	NUM
ap-7678	531	3	]	]	PUNCT
ap-7678	531	4	,	,	PUNCT
ap-7678	531	5	and	and	CCONJ
ap-7678	531	6	theoretically	theoretically	ADV
ap-7678	531	7	confirmed	confirm	VERB
ap-7678	531	8	by	by	ADP
ap-7678	531	9	dorey	dorey	PROPN
ap-7678	531	10	et	et	PROPN
ap-7678	531	11	al	al	PROPN
ap-7678	531	12	.	.	PUNCT
ap-7678	532	1	[	[	X
ap-7678	532	2	6	6	NUM
ap-7678	532	3	]	]	PUNCT
ap-7678	532	4	.	.	PUNCT
ap-7678	533	1	the	the	DET
ap-7678	533	2	emm	emm	PROPN
ap-7678	533	3	-	-	PUNCT
ap-7678	533	4	ψ∗ψ	ψ∗ψ	PUNCT
ap-7678	533	5	formulation	formulation	NOUN
ap-7678	533	6	also	also	ADV
ap-7678	533	7	provided	provide	VERB
ap-7678	533	8	strong	strong	ADJ
ap-7678	533	9	numerical	numerical	ADJ
ap-7678	533	10	evidence	evidence	NOUN
ap-7678	533	11	for	for	ADP
ap-7678	533	12	this	this	PRON
ap-7678	533	13	[	[	X
ap-7678	533	14	27	27	NUM
ap-7678	533	15	]	]	PUNCT
ap-7678	533	16	.	.	PUNCT
ap-7678	534	1	states	state	NOUN
ap-7678	534	2	that	that	PRON
ap-7678	534	3	are	be	AUX
ap-7678	534	4	pt	pt	X
ap-7678	534	5	symmetric	symmetric	ADJ
ap-7678	534	6	satisfy	satisfy	NOUN
ap-7678	534	7	ψ∗(−x	ψ∗(−x	NOUN
ap-7678	534	8	)	)	PUNCT
ap-7678	534	9	=	=	SYM
ap-7678	534	10	ψ(x	ψ(x	NOUN
ap-7678	534	11	)	)	PUNCT
ap-7678	534	12	,	,	PUNCT
ap-7678	534	13	and	and	CCONJ
ap-7678	534	14	have	have	VERB
ap-7678	534	15	real	real	ADJ
ap-7678	534	16	energies	energy	NOUN
ap-7678	534	17	,	,	PUNCT
ap-7678	534	18	e	e	PROPN
ap-7678	534	19	∈	∈	PROPN
ap-7678	534	20	ℜ.	ℜ.	PROPN
ap-7678	534	21	states	state	NOUN
ap-7678	534	22	that	that	PRON
ap-7678	534	23	break	break	VERB
ap-7678	534	24	this	this	DET
ap-7678	534	25	symmetry	symmetry	NOUN
ap-7678	534	26	have	have	VERB
ap-7678	534	27	complex	complex	ADJ
ap-7678	534	28	conjugate	conjugate	ADJ
ap-7678	534	29	pairs	pair	NOUN
ap-7678	534	30	for	for	ADP
ap-7678	534	31	the	the	DET
ap-7678	534	32	energies	energy	NOUN
ap-7678	534	33	(	(	PUNCT
ap-7678	534	34	i.e.	i.e.	X
ap-7678	534	35	ψ∗(−x	ψ∗(−x	ADJ
ap-7678	534	36	)	)	PUNCT
ap-7678	534	37	̸=	̸=	PROPN
ap-7678	534	38	ψ(x	ψ(x	NUM
ap-7678	534	39	)	)	PUNCT
ap-7678	534	40	,	,	PUNCT
ap-7678	534	41	with	with	ADP
ap-7678	534	42	complex	complex	ADJ
ap-7678	534	43	energies	energy	NOUN
ap-7678	534	44	e∗	e∗	PROPN
ap-7678	534	45	and	and	CCONJ
ap-7678	534	46	e	e	NOUN
ap-7678	534	47	,	,	PUNCT
ap-7678	534	48	respectively	respectively	ADV
ap-7678	534	49	)	)	PUNCT
ap-7678	534	50	.	.	PUNCT
ap-7678	535	1	we	we	PRON
ap-7678	535	2	work	work	VERB
ap-7678	535	3	with	with	ADP
ap-7678	535	4	the	the	DET
ap-7678	535	5	mer	mer	PROPN
ap-7678	535	6	-	-	PROPN
ap-7678	535	7	ψ	ψ	NOUN
ap-7678	535	8	representation	representation	NOUN
ap-7678	535	9	for	for	ADP
ap-7678	535	10	which	which	PRON
ap-7678	535	11	the	the	DET
ap-7678	535	12	complex	complex	ADJ
ap-7678	535	13	power	power	NOUN
ap-7678	535	14	moments	moment	NOUN
ap-7678	535	15	on	on	ADP
ap-7678	535	16	the	the	DET
ap-7678	535	17	real	real	ADJ
ap-7678	535	18	axis	axis	NOUN
ap-7678	535	19	satisfy	satisfy	NOUN
ap-7678	535	20	µ(p	µ(p	PROPN
ap-7678	536	1	+	+	CCONJ
ap-7678	537	1	3	3	X
ap-7678	537	2	)	)	PUNCT
ap-7678	537	3	=	=	SYM
ap-7678	537	4	−	−	NOUN
ap-7678	537	5	aµ(p	aµ(p	NOUN
ap-7678	537	6	+	+	CCONJ
ap-7678	537	7	1	1	X
ap-7678	537	8	)	)	PUNCT
ap-7678	537	9	−	−	PROPN
ap-7678	537	10	ieµ(p	ieµ(p	NOUN
ap-7678	537	11	)	)	PUNCT
ap-7678	537	12	−	−	NOUN
ap-7678	537	13	ip(p	ip(p	NUM
ap-7678	538	1	−	−	PROPN
ap-7678	538	2	1)µ(p	1)µ(p	NUM
ap-7678	538	3	−	−	NOUN
ap-7678	538	4	2	2	NUM
ap-7678	538	5	)	)	PUNCT
ap-7678	538	6	,	,	PUNCT
ap-7678	538	7	p	p	PRON
ap-7678	538	8	≥	≥	NOUN
ap-7678	538	9	0	0	NUM
ap-7678	538	10	.	.	PUNCT
ap-7678	539	1	(	(	PUNCT
ap-7678	539	2	72	72	NUM
ap-7678	539	3	)	)	PUNCT
ap-7678	539	4	the	the	DET
ap-7678	539	5	recursion	recursion	NOUN
ap-7678	539	6	relation	relation	NOUN
ap-7678	539	7	for	for	ADP
ap-7678	539	8	the	the	DET
ap-7678	539	9	energy	energy	NOUN
ap-7678	539	10	dependent	dependent	ADJ
ap-7678	539	11	generator	generator	NOUN
ap-7678	539	12	coefficients	coefficient	NOUN
ap-7678	539	13	is	be	AUX
ap-7678	539	14	me(p	me(p	ADJ
ap-7678	539	15	+	+	CCONJ
ap-7678	539	16	3	3	NUM
ap-7678	539	17	,	,	PUNCT
ap-7678	539	18	ℓ	ℓ	NUM
ap-7678	539	19	)	)	PUNCT
ap-7678	539	20	=	=	PUNCT
ap-7678	540	1	−	−	PROPN
ap-7678	540	2	ame(p	ame(p	PROPN
ap-7678	540	3	+	+	PROPN
ap-7678	540	4	1	1	NUM
ap-7678	540	5	,	,	PUNCT
ap-7678	540	6	ℓ	ℓ	NUM
ap-7678	540	7	)	)	PUNCT
ap-7678	540	8	−	−	PROPN
ap-7678	540	9	ieme(p	ieme(p	PROPN
ap-7678	540	10	,	,	PUNCT
ap-7678	540	11	ℓ	ℓ	NUM
ap-7678	540	12	)	)	PUNCT
ap-7678	540	13	−	−	NOUN
ap-7678	540	14	ip(p	ip(p	NUM
ap-7678	540	15	−	−	PROPN
ap-7678	540	16	1)me(p	1)me(p	NUM
ap-7678	540	17	−	−	PROPN
ap-7678	540	18	2	2	NUM
ap-7678	540	19	,	,	PUNCT
ap-7678	540	20	ℓ	ℓ	NUM
ap-7678	540	21	)	)	PUNCT
ap-7678	540	22	,	,	PUNCT
ap-7678	540	23	(	(	PUNCT
ap-7678	540	24	73	73	NUM
ap-7678	540	25	)	)	PUNCT
ap-7678	540	26	for	for	ADP
ap-7678	540	27	p	p	PRON
ap-7678	540	28	≥	≥	NOUN
ap-7678	540	29	0	0	NUM
ap-7678	540	30	,	,	PUNCT
ap-7678	540	31	0	0	NUM
ap-7678	540	32	≤	≤	NUM
ap-7678	540	33	ℓ	ℓ	NOUN
ap-7678	540	34	≤	≤	NUM
ap-7678	540	35	2	2	NUM
ap-7678	540	36	,	,	PUNCT
ap-7678	540	37	and	and	CCONJ
ap-7678	540	38	me(ℓ1	me(ℓ1	NOUN
ap-7678	540	39	,	,	PUNCT
ap-7678	540	40	ℓ2	ℓ2	NOUN
ap-7678	540	41	)	)	PUNCT
ap-7678	540	42	=	=	PUNCT
ap-7678	540	43	δℓ1,ℓ2	δℓ1,ℓ2	ADJ
ap-7678	540	44	.	.	PUNCT
ap-7678	541	1	this	this	PRON
ap-7678	541	2	is	be	AUX
ap-7678	541	3	an	an	DET
ap-7678	541	4	ms	ms	NOUN
ap-7678	541	5	=	=	SYM
ap-7678	541	6	2	2	NUM
ap-7678	541	7	representation	representation	NOUN
ap-7678	541	8	of	of	ADP
ap-7678	541	9	order	order	NOUN
ap-7678	541	10	3	3	X
ap-7678	541	11	.	.	PUNCT
ap-7678	542	1	since	since	SCONJ
ap-7678	542	2	the	the	DET
ap-7678	542	3	asymptotic	asymptotic	ADJ
ap-7678	542	4	form	form	NOUN
ap-7678	542	5	of	of	ADP
ap-7678	542	6	the	the	DET
ap-7678	542	7	physical	physical	ADJ
ap-7678	542	8	states	state	NOUN
ap-7678	542	9	goes	go	VERB
ap-7678	542	10	as	as	ADP
ap-7678	542	11	ψ(x	ψ(x	NOUN
ap-7678	542	12	)	)	PUNCT
ap-7678	542	13	∼	∼	NOUN
ap-7678	542	14	exp(−	exp(−	ADJ
ap-7678	542	15	2	2	NUM
ap-7678	542	16	5	5	NUM
ap-7678	542	17	|x|	|x|	PROPN
ap-7678	542	18	5	5	NUM
ap-7678	542	19	2	2	NUM
ap-7678	542	20	)	)	PUNCT
ap-7678	542	21	,	,	PUNCT
ap-7678	542	22	we	we	PRON
ap-7678	542	23	can	can	AUX
ap-7678	542	24	use	use	VERB
ap-7678	542	25	the	the	DET
ap-7678	542	26	gaussian	gaussian	ADJ
ap-7678	542	27	weight	weight	NOUN
ap-7678	542	28	r(x	r(x	PROPN
ap-7678	542	29	)	)	PUNCT
ap-7678	542	30	=	=	SYM
ap-7678	542	31	exp(−x2/2	exp(−x2/2	X
ap-7678	542	32	)	)	PUNCT
ap-7678	542	33	.	.	PUNCT
ap-7678	543	1	the	the	DET
ap-7678	543	2	oppq	oppq	PROPN
ap-7678	543	3	-	-	PUNCT
ap-7678	543	4	bm	bm	PROPN
ap-7678	543	5	formalism	formalism	NOUN
ap-7678	543	6	ensues	ensue	VERB
ap-7678	543	7	as	as	ADP
ap-7678	543	8	before	before	ADV
ap-7678	543	9	:	:	PUNCT
ap-7678	543	10	ψ(x	ψ(x	NUM
ap-7678	543	11	)	)	PUNCT
ap-7678	544	1	=	=	PUNCT
ap-7678	545	1	∞∑	∞∑	NUM
ap-7678	545	2	j=0	j=0	PROPN
ap-7678	545	3	cjpj(x)r(x	cjpj(x)r(x	NOUN
ap-7678	545	4	)	)	PUNCT
ap-7678	545	5	,	,	PUNCT
ap-7678	545	6	(	(	PUNCT
ap-7678	545	7	74	74	X
ap-7678	545	8	)	)	PUNCT
ap-7678	545	9	involving	involve	VERB
ap-7678	545	10	real	real	ADJ
ap-7678	545	11	weights	weight	NOUN
ap-7678	545	12	and	and	CCONJ
ap-7678	545	13	corresponding	correspond	VERB
ap-7678	545	14	orthonormal	orthonormal	ADJ
ap-7678	545	15	polynomials	polynomial	NOUN
ap-7678	545	16	.	.	PUNCT
ap-7678	546	1	the	the	DET
ap-7678	546	2	complex	complex	ADJ
ap-7678	546	3	projection	projection	NOUN
ap-7678	546	4	coefficients	coefficient	NOUN
ap-7678	546	5	are	be	AUX
ap-7678	546	6	given	give	VERB
ap-7678	546	7	by	by	ADP
ap-7678	546	8	cj(e	cj(e	NUM
ap-7678	546	9	,	,	PUNCT
ap-7678	546	10	µ0	µ0	NOUN
ap-7678	546	11	,	,	PUNCT
ap-7678	546	12	µ1	µ1	PROPN
ap-7678	546	13	,	,	PUNCT
ap-7678	546	14	µ2	µ2	PROPN
ap-7678	546	15	)	)	PUNCT
ap-7678	546	16	=	=	SYM
ap-7678	546	17	⟨pj(x)|ψ⟩	⟨pj(x)|ψ⟩	PROPN
ap-7678	546	18	,	,	PUNCT
ap-7678	546	19	(	(	PUNCT
ap-7678	546	20	75	75	NUM
ap-7678	546	21	)	)	PUNCT
ap-7678	546	22	or	or	CCONJ
ap-7678	546	23	cj(e	cj(e	NUM
ap-7678	546	24	,	,	PUNCT
ap-7678	546	25	µ0	µ0	PROPN
ap-7678	546	26	,	,	PUNCT
ap-7678	546	27	µ1	µ1	PROPN
ap-7678	546	28	,	,	PUNCT
ap-7678	546	29	µ2	µ2	PROPN
ap-7678	546	30	)	)	PUNCT
ap-7678	546	31	=	=	SYM
ap-7678	546	32	ms=2∑	ms=2∑	X
ap-7678	546	33	ℓ=0	ℓ=0	SYM
ap-7678	546	34	λ(j	λ(j	PROPN
ap-7678	546	35	)	)	PUNCT
ap-7678	546	36	ℓ	ℓ	PROPN
ap-7678	546	37	(	(	PUNCT
ap-7678	546	38	e	e	NOUN
ap-7678	546	39	)	)	PUNCT
ap-7678	546	40	µℓ	µℓ	ADP
ap-7678	546	41	,	,	PUNCT
ap-7678	546	42	(	(	PUNCT
ap-7678	546	43	76	76	NUM
ap-7678	546	44	)	)	PUNCT
ap-7678	546	45	where	where	SCONJ
ap-7678	546	46	the	the	DET
ap-7678	546	47	coefficients	coefficient	NOUN
ap-7678	546	48	depend	depend	VERB
ap-7678	546	49	on	on	ADP
ap-7678	546	50	the	the	DET
ap-7678	546	51	orthonormal	orthonormal	ADJ
ap-7678	546	52	polynomial	polynomial	ADJ
ap-7678	546	53	coefficients	coefficient	NOUN
ap-7678	546	54	and	and	CCONJ
ap-7678	546	55	the	the	DET
ap-7678	546	56	me(p	me(p	ADJ
ap-7678	546	57	,	,	PUNCT
ap-7678	546	58	ℓ	ℓ	NOUN
ap-7678	546	59	)	)	PUNCT
ap-7678	546	60	’s	’s	ADV
ap-7678	546	61	:	:	PUNCT
ap-7678	546	62	λ(j	λ(j	X
ap-7678	546	63	)	)	PUNCT
ap-7678	546	64	ℓ	ℓ	PROPN
ap-7678	546	65	(	(	PUNCT
ap-7678	546	66	e	e	NOUN
ap-7678	546	67	)	)	PUNCT
ap-7678	546	68	=	=	SYM
ap-7678	547	1	j∑	j∑	PROPN
ap-7678	547	2	η=0	η=0	PROPN
ap-7678	547	3	ξ(j	ξ(j	PROPN
ap-7678	547	4	)	)	PUNCT
ap-7678	547	5	η	η	NOUN
ap-7678	547	6	me(η	me(η	X
ap-7678	547	7	,	,	PUNCT
ap-7678	547	8	ℓ	ℓ	NOUN
ap-7678	547	9	)	)	PUNCT
ap-7678	547	10	.	.	PUNCT
ap-7678	548	1	(	(	PUNCT
ap-7678	548	2	77	77	X
ap-7678	548	3	)	)	PUNCT
ap-7678	548	4	the	the	DET
ap-7678	548	5	oppq	oppq	PROPN
ap-7678	548	6	-	-	PUNCT
ap-7678	548	7	bm	bm	PROPN
ap-7678	548	8	quantization	quantization	NOUN
ap-7678	548	9	condition	condition	NOUN
ap-7678	548	10	is	be	AUX
ap-7678	548	11	i[ψ	i[ψ	NOUN
ap-7678	548	12	,	,	PUNCT
ap-7678	548	13	r	r	X
ap-7678	548	14	]	]	PUNCT
ap-7678	548	15	=	=	PUNCT
ap-7678	548	16	∫	∫	PROPN
ap-7678	548	17	ℜ	ℜ	PROPN
ap-7678	548	18	dx	dx	PROPN
ap-7678	548	19	ψ∗(x)ψ(x	ψ∗(x)ψ(x	PROPN
ap-7678	548	20	)	)	PUNCT
ap-7678	548	21	r(x	r(x	PROPN
ap-7678	548	22	)	)	PUNCT
ap-7678	549	1	=	=	PRON
ap-7678	549	2	{	{	PUNCT
ap-7678	549	3	finite	finite	VERB
ap-7678	549	4	⇐	⇐	ADJ
ap-7678	549	5	⇒	⇒	NOUN
ap-7678	549	6	e	e	NOUN
ap-7678	549	7	=	=	SYM
ap-7678	549	8	ephys	ephy	NOUN
ap-7678	549	9	and	and	CCONJ
ap-7678	549	10	−→µ	−→µ	NOUN
ap-7678	549	11	=	=	SYM
ap-7678	549	12	−→µ	−→µ	NOUN
ap-7678	549	13	phys	phy	NOUN
ap-7678	549	14	,	,	PUNCT
ap-7678	550	1	∞	∞	NUM
ap-7678	550	2	⇐	⇐	ADJ
ap-7678	550	3	⇒	⇒	NOUN
ap-7678	550	4	e	e	X
ap-7678	550	5	̸=	̸=	PROPN
ap-7678	550	6	ephys	ephy	NOUN
ap-7678	550	7	or	or	CCONJ
ap-7678	550	8	−→µ	−→µ	NOUN
ap-7678	550	9	̸=	̸=	PROPN
ap-7678	550	10	−→µ	−→µ	NOUN
ap-7678	550	11	phys	phy	NOUN
ap-7678	550	12	.	.	PUNCT
ap-7678	551	1	(	(	PUNCT
ap-7678	551	2	78	78	X
ap-7678	551	3	)	)	PUNCT
ap-7678	551	4	we	we	PRON
ap-7678	551	5	adopt	adopt	VERB
ap-7678	551	6	the	the	DET
ap-7678	551	7	vector	vector	NOUN
ap-7678	551	8	notation	notation	NOUN
ap-7678	551	9	for	for	ADP
ap-7678	551	10	the	the	DET
ap-7678	551	11	missing	miss	VERB
ap-7678	551	12	moments	moment	NOUN
ap-7678	551	13	−→µ	−→µ	PROPN
ap-7678	551	14	≡	≡	PROPN
ap-7678	551	15	(	(	PUNCT
ap-7678	551	16	µ0	µ0	PROPN
ap-7678	551	17	,	,	PUNCT
ap-7678	551	18	µ1	µ1	PROPN
ap-7678	551	19	,	,	PUNCT
ap-7678	551	20	µ2	µ2	PROPN
ap-7678	551	21	)	)	PUNCT
ap-7678	551	22	.	.	PUNCT
ap-7678	552	1	substituting	substitute	VERB
ap-7678	552	2	the	the	DET
ap-7678	552	3	oppq	oppq	ADJ
ap-7678	552	4	representation	representation	NOUN
ap-7678	552	5	we	we	PRON
ap-7678	552	6	obtain	obtain	VERB
ap-7678	552	7	:	:	PUNCT
ap-7678	552	8	i[ψ	i[ψ	NOUN
ap-7678	552	9	,	,	PUNCT
ap-7678	552	10	r	r	NOUN
ap-7678	552	11	]	]	X
ap-7678	552	12	=	=	SYM
ap-7678	553	1	∞∑	∞∑	NUM
ap-7678	553	2	j=0	j=0	PROPN
ap-7678	553	3	c∗	c∗	PROPN
ap-7678	553	4	j	j	PROPN
ap-7678	553	5	(	(	PUNCT
ap-7678	553	6	e	e	NOUN
ap-7678	553	7	,	,	PUNCT
ap-7678	553	8	−→µ	−→µ	NOUN
ap-7678	553	9	)	)	PUNCT
ap-7678	553	10	cj(e	cj(e	PUNCT
ap-7678	553	11	,	,	PUNCT
ap-7678	553	12	−→µ	−→µ	NOUN
ap-7678	553	13	)	)	PUNCT
ap-7678	553	14	.	.	PUNCT
ap-7678	554	1	(	(	PUNCT
ap-7678	554	2	79	79	NUM
ap-7678	554	3	)	)	PUNCT
ap-7678	554	4	once	once	ADV
ap-7678	554	5	again	again	ADV
ap-7678	554	6	,	,	PUNCT
ap-7678	554	7	the	the	DET
ap-7678	554	8	focus	focus	NOUN
ap-7678	554	9	is	be	AUX
ap-7678	554	10	on	on	ADP
ap-7678	554	11	:	:	PUNCT
ap-7678	554	12	sn	sn	PROPN
ap-7678	554	13	(	(	PUNCT
ap-7678	554	14	e	e	NOUN
ap-7678	554	15	,	,	PUNCT
ap-7678	554	16	−→µ	−→µ	NOUN
ap-7678	554	17	)	)	PUNCT
ap-7678	554	18	=	=	SYM
ap-7678	554	19	ms∑	ms∑	AUX
ap-7678	554	20	ℓ1=0	ℓ1=0	PROPN
ap-7678	554	21	ms∑	ms∑	PROPN
ap-7678	554	22	ℓ2=0	ℓ2=0	NOUN
ap-7678	554	23	µ∗	µ∗	VERB
ap-7678	554	24	ℓ1	ℓ1	NOUN
ap-7678	554	25	(	(	PUNCT
ap-7678	554	26	n∑	n∑	X
ap-7678	554	27	j=0	j=0	PROPN
ap-7678	554	28	λ(j	λ(j	PROPN
ap-7678	554	29	)	)	PUNCT
ap-7678	554	30	ℓ1	ℓ1	NOUN
ap-7678	554	31	(	(	PUNCT
ap-7678	554	32	e)∗λ(j	e)∗λ(j	NOUN
ap-7678	554	33	)	)	PUNCT
ap-7678	554	34	ℓ2	ℓ2	NOUN
ap-7678	554	35	(	(	PUNCT
ap-7678	554	36	e	e	NOUN
ap-7678	554	37	)	)	PUNCT
ap-7678	554	38	)	)	PUNCT
ap-7678	554	39	µℓ2	µℓ2	ADV
ap-7678	554	40	,	,	PUNCT
ap-7678	554	41	(	(	PUNCT
ap-7678	554	42	80	80	NUM
ap-7678	554	43	)	)	PUNCT
ap-7678	554	44	involving	involve	VERB
ap-7678	554	45	the	the	DET
ap-7678	554	46	positive	positive	ADJ
ap-7678	554	47	definite	definite	ADJ
ap-7678	554	48	matrix	matrix	NOUN
ap-7678	554	49	(	(	PUNCT
ap-7678	554	50	pn	pn	X
ap-7678	554	51	(	(	PUNCT
ap-7678	554	52	e	e	NOUN
ap-7678	554	53	)	)	PUNCT
ap-7678	554	54	)	)	PUNCT
ap-7678	555	1	ℓ1,ℓ2	ℓ1,ℓ2	PROPN
ap-7678	555	2	=	=	SYM
ap-7678	555	3	n∑	n∑	PROPN
ap-7678	555	4	j=0	j=0	PROPN
ap-7678	555	5	λ(j	λ(j	PROPN
ap-7678	555	6	)	)	PUNCT
ap-7678	555	7	ℓ1	ℓ1	NOUN
ap-7678	555	8	(	(	PUNCT
ap-7678	555	9	e)∗λ(j	e)∗λ(j	NOUN
ap-7678	555	10	)	)	PUNCT
ap-7678	555	11	ℓ2	ℓ2	NOUN
ap-7678	555	12	(	(	PUNCT
ap-7678	555	13	e	e	NOUN
ap-7678	555	14	)	)	PUNCT
ap-7678	555	15	.	.	PUNCT
ap-7678	556	1	(	(	PUNCT
ap-7678	556	2	81	81	NUM
ap-7678	556	3	)	)	PUNCT
ap-7678	556	4	if	if	SCONJ
ap-7678	556	5	we	we	PRON
ap-7678	556	6	adopt	adopt	VERB
ap-7678	556	7	a	a	DET
ap-7678	556	8	unit	unit	NOUN
ap-7678	556	9	missing	miss	VERB
ap-7678	556	10	moment	moment	NOUN
ap-7678	556	11	vector	vector	NOUN
ap-7678	556	12	normalization	normalization	NOUN
ap-7678	556	13	,	,	PUNCT
ap-7678	556	14	then	then	ADV
ap-7678	556	15	the	the	DET
ap-7678	556	16	focus	focus	NOUN
ap-7678	556	17	for	for	ADP
ap-7678	556	18	oppq	oppq	NOUN
ap-7678	556	19	-	-	PUNCT
ap-7678	556	20	bm	bm	PROPN
ap-7678	556	21	quantization	quantization	NOUN
ap-7678	556	22	is	be	AUX
ap-7678	556	23	on	on	ADP
ap-7678	556	24	the	the	DET
ap-7678	556	25	behavior	behavior	NOUN
ap-7678	556	26	of	of	ADP
ap-7678	556	27	the	the	DET
ap-7678	556	28	smallest	small	ADJ
ap-7678	556	29	eigenvalue	eigenvalue	NOUN
ap-7678	556	30	,	,	PUNCT
ap-7678	556	31	λn	λn	X
ap-7678	556	32	(	(	PUNCT
ap-7678	556	33	e	e	NOUN
ap-7678	556	34	)	)	PUNCT
ap-7678	556	35	≡	≡	PROPN
ap-7678	556	36	smallest	small	ADJ
ap-7678	556	37	eigenvalue	eigenvalue	NOUN
ap-7678	556	38	(	(	PUNCT
ap-7678	556	39	pn	pn	PROPN
ap-7678	556	40	(	(	PUNCT
ap-7678	556	41	e	e	NOUN
ap-7678	556	42	)	)	PUNCT
ap-7678	556	43	)	)	PUNCT
ap-7678	556	44	,	,	PUNCT
ap-7678	556	45	viewed	view	VERB
ap-7678	556	46	as	as	ADP
ap-7678	556	47	a	a	DET
ap-7678	556	48	function	function	NOUN
ap-7678	556	49	of	of	ADP
ap-7678	556	50	the	the	DET
ap-7678	556	51	real	real	ADJ
ap-7678	556	52	and	and	CCONJ
ap-7678	556	53	imaginary	imaginary	ADJ
ap-7678	556	54	parts	part	NOUN
ap-7678	556	55	of	of	ADP
ap-7678	556	56	the	the	DET
ap-7678	556	57	energy	energy	NOUN
ap-7678	556	58	variable	variable	NOUN
ap-7678	556	59	.	.	PUNCT
ap-7678	557	1	the	the	DET
ap-7678	557	2	focus	focus	NOUN
ap-7678	557	3	is	be	AUX
ap-7678	557	4	on	on	ADP
ap-7678	557	5	determining	determine	VERB
ap-7678	557	6	the	the	DET
ap-7678	557	7	local	local	ADJ
ap-7678	557	8	minima	minima	NOUN
ap-7678	557	9	:	:	PUNCT
ap-7678	557	10	∂er	∂er	NOUN
ap-7678	557	11	λn	λn	NOUN
ap-7678	557	12	(	(	PUNCT
ap-7678	557	13	e(min	e(min	PROPN
ap-7678	557	14	)	)	PUNCT
ap-7678	557	15	n	n	CCONJ
ap-7678	557	16	,	,	PUNCT
ap-7678	557	17	phys	phy	NOUN
ap-7678	557	18	)	)	PUNCT
ap-7678	557	19	=	=	SYM
ap-7678	557	20	∂ei	∂ei	PRON
ap-7678	557	21	λn	λn	NOUN
ap-7678	557	22	(	(	PUNCT
ap-7678	557	23	e(min	e(min	PROPN
ap-7678	557	24	)	)	PUNCT
ap-7678	557	25	n	n	CCONJ
ap-7678	557	26	,	,	PUNCT
ap-7678	557	27	phys	phy	NOUN
ap-7678	557	28	)	)	PUNCT
ap-7678	557	29	=	=	SYM
ap-7678	558	1	0	0	X
ap-7678	558	2	.	.	PUNCT
ap-7678	559	1	these	these	DET
ap-7678	559	2	derivatives	derivative	NOUN
ap-7678	559	3	can	can	AUX
ap-7678	559	4	be	be	AUX
ap-7678	559	5	obtained	obtain	VERB
ap-7678	559	6	algebraically	algebraically	ADV
ap-7678	559	7	.	.	PUNCT
ap-7678	560	1	we	we	PRON
ap-7678	560	2	outline	outline	VERB
ap-7678	560	3	the	the	DET
ap-7678	560	4	essential	essential	ADJ
ap-7678	560	5	steps	step	NOUN
ap-7678	560	6	for	for	ADP
ap-7678	560	7	algebraically	algebraically	ADV
ap-7678	560	8	generating	generate	VERB
ap-7678	560	9	the	the	DET
ap-7678	560	10	partial	partial	ADJ
ap-7678	560	11	derivatives	derivative	NOUN
ap-7678	560	12	.	.	PUNCT
ap-7678	561	1	first	first	ADV
ap-7678	561	2	of	of	ADP
ap-7678	561	3	all	all	PRON
ap-7678	561	4	,	,	PUNCT
ap-7678	561	5	we	we	PRON
ap-7678	561	6	have	have	VERB
ap-7678	561	7	λn	λn	NOUN
ap-7678	561	8	(	(	PUNCT
ap-7678	561	9	e	e	NOUN
ap-7678	561	10	)	)	PUNCT
ap-7678	561	11	=	=	SYM
ap-7678	561	12	⟨−→µ	⟨−→µ	NOUN
ap-7678	561	13	|pn	|pn	X
ap-7678	561	14	(	(	PUNCT
ap-7678	561	15	e)|−→µ	e)|−→µ	NOUN
ap-7678	561	16	⟩	⟩	NOUN
ap-7678	561	17	,	,	PUNCT
ap-7678	561	18	involving	involve	VERB
ap-7678	561	19	the	the	DET
ap-7678	561	20	normalized	normalize	VERB
ap-7678	561	21	missing	missing	ADJ
ap-7678	561	22	moment	moment	NOUN
ap-7678	561	23	(	(	PUNCT
ap-7678	561	24	lowest	low	ADJ
ap-7678	561	25	)	)	PUNCT
ap-7678	561	26	eigenvector	eigenvector	NOUN
ap-7678	561	27	.	.	PUNCT
ap-7678	562	1	this	this	PRON
ap-7678	562	2	is	be	AUX
ap-7678	562	3	not	not	PART
ap-7678	562	4	an	an	DET
ap-7678	562	5	analytic	analytic	ADJ
ap-7678	562	6	function	function	NOUN
ap-7678	562	7	in	in	ADP
ap-7678	562	8	e	e	NOUN
ap-7678	562	9	,	,	PUNCT
ap-7678	562	10	so	so	SCONJ
ap-7678	562	11	we	we	PRON
ap-7678	562	12	must	must	AUX
ap-7678	562	13	work	work	VERB
ap-7678	562	14	with	with	ADP
ap-7678	562	15	the	the	DET
ap-7678	562	16	partial	partial	ADJ
ap-7678	562	17	derivatives	derivative	NOUN
ap-7678	562	18	with	with	ADP
ap-7678	562	19	respects	respect	NOUN
ap-7678	562	20	to	to	ADP
ap-7678	562	21	er	er	INTJ
ap-7678	562	22	,	,	PUNCT
ap-7678	562	23	i.	i.	PROPN
ap-7678	562	24	thus	thus	ADV
ap-7678	562	25	,	,	PUNCT
ap-7678	562	26	we	we	PRON
ap-7678	562	27	need	need	VERB
ap-7678	562	28	∂er	∂er	NOUN
ap-7678	562	29	,	,	PUNCT
ap-7678	562	30	iλn	iλn	X
ap-7678	562	31	(	(	PUNCT
ap-7678	562	32	e	e	NOUN
ap-7678	562	33	)	)	PUNCT
ap-7678	562	34	=	=	SYM
ap-7678	562	35	⟨−→µ	⟨−→µ	NOUN
ap-7678	562	36	|∂er	|∂er	NOUN
ap-7678	562	37	,	,	PUNCT
ap-7678	562	38	ipn	ipn	PROPN
ap-7678	562	39	(	(	PUNCT
ap-7678	562	40	e)|−→µ	e)|−→µ	NOUN
ap-7678	562	41	⟩.	⟩.	NOUN
ap-7678	562	42	we	we	PRON
ap-7678	562	43	can	can	AUX
ap-7678	562	44	generate	generate	VERB
ap-7678	562	45	the	the	DET
ap-7678	562	46	required	require	VERB
ap-7678	562	47	matrix	matrix	NOUN
ap-7678	562	48	expression	expression	NOUN
ap-7678	562	49	from	from	ADP
ap-7678	562	50	eq	eq	ADP
ap-7678	562	51	.	.	PUNCT
ap-7678	563	1	(	(	PUNCT
ap-7678	563	2	80	80	NUM
ap-7678	563	3	)	)	PUNCT
ap-7678	563	4	or	or	CCONJ
ap-7678	563	5	∂er	∂er	NOUN
ap-7678	563	6	,	,	PUNCT
ap-7678	563	7	i	i	PRON
ap-7678	563	8	pn	pn	X
ap-7678	563	9	(	(	PUNCT
ap-7678	563	10	e	e	NOUN
ap-7678	563	11	)	)	PUNCT
ap-7678	563	12	=	=	SYM
ap-7678	564	1	∑n	∑n	PROPN
ap-7678	564	2	j=0	j=0	PROPN
ap-7678	564	3	(	(	PUNCT
ap-7678	564	4	∂er	∂er	NOUN
ap-7678	564	5	,	,	PUNCT
ap-7678	564	6	i	i	PRON
ap-7678	564	7	λ(j)(e)∗	λ(j)(e)∗	PUNCT
ap-7678	564	8	)	)	PUNCT
ap-7678	564	9	λ(j)(e	λ(j)(e	NUM
ap-7678	564	10	)	)	PUNCT
ap-7678	564	11	+	+	CCONJ
ap-7678	564	12	c.c	c.c	PROPN
ap-7678	564	13	..	..	PROPN
ap-7678	564	14	from	from	ADP
ap-7678	564	15	eq	eq	PROPN
ap-7678	564	16	.	.	PUNCT
ap-7678	565	1	(	(	PUNCT
ap-7678	565	2	77	77	NUM
ap-7678	565	3	)	)	PUNCT
ap-7678	565	4	,	,	PUNCT
ap-7678	565	5	so	so	ADV
ap-7678	565	6	long	long	ADV
ap-7678	565	7	as	as	SCONJ
ap-7678	565	8	the	the	DET
ap-7678	565	9	ξ	ξ	PROPN
ap-7678	565	10	’s	’s	PART
ap-7678	565	11	do	do	AUX
ap-7678	565	12	not	not	PART
ap-7678	565	13	depend	depend	VERB
ap-7678	565	14	on	on	ADP
ap-7678	565	15	the	the	DET
ap-7678	565	16	energy	energy	NOUN
ap-7678	565	17	parameter	parameter	NOUN
ap-7678	565	18	,	,	PUNCT
ap-7678	565	19	we	we	PRON
ap-7678	565	20	need	need	VERB
ap-7678	565	21	∂er	∂er	NOUN
ap-7678	565	22	,	,	PUNCT
ap-7678	565	23	i	i	PRON
ap-7678	565	24	me(η	me(η	VERB
ap-7678	565	25	,	,	PUNCT
ap-7678	565	26	ℓ	ℓ	NOUN
ap-7678	565	27	)	)	PUNCT
ap-7678	565	28	.	.	PUNCT
ap-7678	566	1	however	however	ADV
ap-7678	566	2	,	,	PUNCT
ap-7678	566	3	this	this	PRON
ap-7678	566	4	can	can	AUX
ap-7678	566	5	be	be	AUX
ap-7678	566	6	obtained	obtain	VERB
ap-7678	566	7	recursively	recursively	ADV
ap-7678	566	8	through	through	ADP
ap-7678	566	9	the	the	DET
ap-7678	566	10	expression	expression	NOUN
ap-7678	566	11	in	in	ADP
ap-7678	566	12	eq	eq	ADP
ap-7678	566	13	.	.	PUNCT
ap-7678	567	1	(	(	PUNCT
ap-7678	567	2	72	72	NUM
ap-7678	567	3	)	)	PUNCT
ap-7678	567	4	.	.	PUNCT
ap-7678	568	1	our	our	PRON
ap-7678	568	2	immediate	immediate	ADJ
ap-7678	568	3	interest	interest	NOUN
ap-7678	568	4	is	be	AUX
ap-7678	568	5	on	on	ADP
ap-7678	568	6	the	the	DET
ap-7678	568	7	behavior	behavior	NOUN
ap-7678	568	8	of	of	ADP
ap-7678	568	9	λn	λn	PROPN
ap-7678	568	10	(	(	PUNCT
ap-7678	568	11	e	e	NOUN
ap-7678	568	12	)	)	PUNCT
ap-7678	568	13	over	over	ADP
ap-7678	568	14	the	the	DET
ap-7678	568	15	two	two	NUM
ap-7678	568	16	dimensional	dimensional	ADJ
ap-7678	568	17	complex	complex	ADJ
ap-7678	568	18	energy	energy	NOUN
ap-7678	568	19	plane	plane	NOUN
ap-7678	568	20	,	,	PUNCT
ap-7678	568	21	as	as	SCONJ
ap-7678	568	22	the	the	DET
ap-7678	568	23	parameter	parameter	NOUN
ap-7678	568	24	‘	'	PUNCT
ap-7678	568	25	a	a	PRON
ap-7678	568	26	’	'	PUNCT
ap-7678	568	27	is	be	AUX
ap-7678	568	28	varied	varied	ADJ
ap-7678	568	29	,	,	PUNCT
ap-7678	568	30	and	and	CCONJ
ap-7678	568	31	n	n	CCONJ
ap-7678	568	32	=	=	NUM
ap-7678	568	33	40	40	NUM
ap-7678	568	34	.	.	PUNCT
ap-7678	569	1	in	in	ADP
ap-7678	569	2	figures	figure	NOUN
ap-7678	569	3	5	5	NUM
ap-7678	569	4	-	-	SYM
ap-7678	569	5	12	12	NUM
ap-7678	569	6	we	we	PRON
ap-7678	569	7	show	show	VERB
ap-7678	569	8	the	the	DET
ap-7678	569	9	convergence	convergence	NOUN
ap-7678	569	10	of	of	ADP
ap-7678	569	11	two	two	NUM
ap-7678	569	12	ptbreaking	ptbreaking	NOUN
ap-7678	569	13	(	(	PUNCT
ap-7678	569	14	complex	complex	ADJ
ap-7678	569	15	conjugate	conjugate	NOUN
ap-7678	569	16	)	)	PUNCT
ap-7678	569	17	energies	energy	NOUN
ap-7678	569	18	(	(	PUNCT
ap-7678	569	19	eventually	eventually	ADV
ap-7678	569	20	becoming	become	VERB
ap-7678	569	21	real	real	ADJ
ap-7678	569	22	)	)	PUNCT
ap-7678	569	23	as	as	SCONJ
ap-7678	569	24	we	we	PRON
ap-7678	569	25	increase	increase	VERB
ap-7678	569	26	the	the	DET
ap-7678	569	27	a	a	DET
ap-7678	569	28	-	-	PUNCT
ap-7678	569	29	parameter	parameter	NOUN
ap-7678	569	30	through	through	ADP
ap-7678	569	31	76	76	NUM
ap-7678	569	32	vol	vol	NOUN
ap-7678	569	33	.	.	PUNCT
ap-7678	570	1	62	62	NUM
ap-7678	570	2	no	no	INTJ
ap-7678	570	3	.	.	PUNCT
ap-7678	571	1	1/2022	1/2022	NUM
ap-7678	571	2	algebraic	algebraic	ADJ
ap-7678	571	3	eigenenergy	eigenenergy	NOUN
ap-7678	571	4	bounding	bounding	NOUN
ap-7678	571	5	method	method	NOUN
ap-7678	571	6	figure	figure	NOUN
ap-7678	571	7	5	5	NUM
ap-7678	571	8	.	.	PUNCT
ap-7678	571	9	log10(λ40(e	log10(λ40(e	NUM
ap-7678	571	10	)	)	PUNCT
ap-7678	571	11	)	)	PUNCT
ap-7678	571	12	for	for	ADP
ap-7678	571	13	pt	pt	NOUN
ap-7678	571	14	-	-	PUNCT
ap-7678	571	15	breaking	break	VERB
ap-7678	571	16	regime	regime	NOUN
ap-7678	571	17	,	,	PUNCT
ap-7678	571	18	−3.30	−3.30	VERB
ap-7678	571	19	≤	≤	NUM
ap-7678	571	20	a	a	DET
ap-7678	571	21	≤	≤	ADJ
ap-7678	571	22	−1.70	−1.70	PRON
ap-7678	571	23	for	for	ADP
ap-7678	571	24	v	v	NOUN
ap-7678	571	25	(	(	PUNCT
ap-7678	571	26	x	x	NOUN
ap-7678	571	27	)	)	PUNCT
ap-7678	571	28	=	=	SYM
ap-7678	571	29	ix3	ix3	PROPN
ap-7678	571	30	+	+	CCONJ
ap-7678	571	31	iax	iax	PROPN
ap-7678	571	32	,	,	PUNCT
ap-7678	571	33	a	a	DET
ap-7678	571	34	=	=	X
ap-7678	571	35	−3.30	−3.30	PROPN
ap-7678	571	36	figure	figure	NOUN
ap-7678	571	37	6	6	NUM
ap-7678	571	38	.	.	PUNCT
ap-7678	571	39	log10(λ40(e	log10(λ40(e	NUM
ap-7678	571	40	)	)	PUNCT
ap-7678	571	41	)	)	PUNCT
ap-7678	571	42	for	for	ADP
ap-7678	571	43	pt	pt	NOUN
ap-7678	571	44	-	-	PUNCT
ap-7678	571	45	breaking	break	VERB
ap-7678	571	46	regime	regime	NOUN
ap-7678	571	47	,	,	PUNCT
ap-7678	571	48	−3.30	−3.30	VERB
ap-7678	571	49	≤	≤	NUM
ap-7678	571	50	a	a	DET
ap-7678	571	51	≤	≤	ADJ
ap-7678	571	52	−1.70	−1.70	PRON
ap-7678	571	53	for	for	ADP
ap-7678	571	54	v	v	NOUN
ap-7678	571	55	(	(	PUNCT
ap-7678	571	56	x	x	NOUN
ap-7678	571	57	)	)	PUNCT
ap-7678	571	58	=	=	SYM
ap-7678	571	59	ix3	ix3	PROPN
ap-7678	571	60	+	+	CCONJ
ap-7678	571	61	iax	iax	PROPN
ap-7678	571	62	,	,	PUNCT
ap-7678	571	63	a	a	DET
ap-7678	571	64	=	=	X
ap-7678	571	65	−2.90	−2.90	NOUN
ap-7678	571	66	figure	figure	NOUN
ap-7678	571	67	7	7	NUM
ap-7678	571	68	.	.	PUNCT
ap-7678	571	69	log10(λ40(e	log10(λ40(e	NUM
ap-7678	571	70	)	)	PUNCT
ap-7678	571	71	)	)	PUNCT
ap-7678	571	72	for	for	ADP
ap-7678	571	73	pt	pt	NOUN
ap-7678	571	74	-	-	PUNCT
ap-7678	571	75	breaking	break	VERB
ap-7678	571	76	regime	regime	NOUN
ap-7678	571	77	,	,	PUNCT
ap-7678	571	78	−3.30	−3.30	VERB
ap-7678	571	79	≤	≤	NUM
ap-7678	571	80	a	a	DET
ap-7678	571	81	≤	≤	ADJ
ap-7678	571	82	−1.70	−1.70	PRON
ap-7678	571	83	for	for	ADP
ap-7678	571	84	v	v	NOUN
ap-7678	571	85	(	(	PUNCT
ap-7678	571	86	x	x	NOUN
ap-7678	571	87	)	)	PUNCT
ap-7678	571	88	=	=	SYM
ap-7678	571	89	ix3	ix3	PROPN
ap-7678	571	90	+	+	CCONJ
ap-7678	571	91	iax	iax	PROPN
ap-7678	571	92	,	,	PUNCT
ap-7678	571	93	a	a	PRON
ap-7678	571	94	=	=	X
ap-7678	571	95	−2.70	−2.70	PRON
ap-7678	571	96	figure	figure	NOUN
ap-7678	571	97	8	8	NUM
ap-7678	571	98	.	.	PUNCT
ap-7678	572	1	log10(λ40(e	log10(λ40(e	NUM
ap-7678	572	2	)	)	PUNCT
ap-7678	572	3	)	)	PUNCT
ap-7678	573	1	for	for	ADP
ap-7678	573	2	pt	pt	NOUN
ap-7678	573	3	-	-	PUNCT
ap-7678	573	4	breaking	break	VERB
ap-7678	573	5	regime	regime	NOUN
ap-7678	573	6	,	,	PUNCT
ap-7678	573	7	−3.30	−3.30	VERB
ap-7678	573	8	≤	≤	NUM
ap-7678	573	9	a	a	DET
ap-7678	573	10	≤	≤	ADJ
ap-7678	573	11	−1.70	−1.70	PRON
ap-7678	573	12	for	for	ADP
ap-7678	573	13	v	v	NOUN
ap-7678	573	14	(	(	PUNCT
ap-7678	573	15	x	x	NOUN
ap-7678	573	16	)	)	PUNCT
ap-7678	573	17	=	=	SYM
ap-7678	573	18	ix3	ix3	PROPN
ap-7678	573	19	+	+	CCONJ
ap-7678	573	20	iax	iax	PROPN
ap-7678	573	21	,	,	PUNCT
ap-7678	573	22	a	a	DET
ap-7678	573	23	=	=	X
ap-7678	573	24	−2.50	−2.50	PROPN
ap-7678	573	25	figure	figure	NOUN
ap-7678	573	26	9	9	NUM
ap-7678	573	27	.	.	PUNCT
ap-7678	573	28	log10(λ40(e	log10(λ40(e	NUM
ap-7678	573	29	)	)	PUNCT
ap-7678	573	30	)	)	PUNCT
ap-7678	573	31	for	for	ADP
ap-7678	573	32	pt	pt	NOUN
ap-7678	573	33	-	-	PUNCT
ap-7678	573	34	breaking	break	VERB
ap-7678	573	35	regime	regime	NOUN
ap-7678	573	36	,	,	PUNCT
ap-7678	573	37	−3.30	−3.30	VERB
ap-7678	573	38	≤	≤	NUM
ap-7678	573	39	a	a	DET
ap-7678	573	40	≤	≤	ADJ
ap-7678	573	41	−1.70	−1.70	PRON
ap-7678	573	42	for	for	ADP
ap-7678	573	43	v	v	NOUN
ap-7678	573	44	(	(	PUNCT
ap-7678	573	45	x	x	NOUN
ap-7678	573	46	)	)	PUNCT
ap-7678	573	47	=	=	SYM
ap-7678	573	48	ix3	ix3	PROPN
ap-7678	573	49	+	+	CCONJ
ap-7678	573	50	iax	iax	PROPN
ap-7678	573	51	,	,	PUNCT
ap-7678	573	52	a	a	DET
ap-7678	573	53	=	=	X
ap-7678	573	54	−2.30	−2.30	ADP
ap-7678	573	55	figure	figure	NOUN
ap-7678	573	56	10	10	NUM
ap-7678	573	57	.	.	PUNCT
ap-7678	574	1	log10(λ40(e	log10(λ40(e	NUM
ap-7678	574	2	)	)	PUNCT
ap-7678	574	3	)	)	PUNCT
ap-7678	575	1	for	for	ADP
ap-7678	575	2	pt	pt	NOUN
ap-7678	575	3	-	-	PUNCT
ap-7678	575	4	breaking	break	VERB
ap-7678	575	5	regime	regime	NOUN
ap-7678	575	6	,	,	PUNCT
ap-7678	575	7	−3.30	−3.30	VERB
ap-7678	575	8	≤	≤	NUM
ap-7678	575	9	a	a	DET
ap-7678	575	10	≤	≤	ADJ
ap-7678	575	11	−1.70	−1.70	PRON
ap-7678	575	12	for	for	ADP
ap-7678	575	13	v	v	NOUN
ap-7678	575	14	(	(	PUNCT
ap-7678	575	15	x	x	NOUN
ap-7678	575	16	)	)	PUNCT
ap-7678	575	17	=	=	SYM
ap-7678	575	18	ix3	ix3	PROPN
ap-7678	575	19	+	+	CCONJ
ap-7678	575	20	iax	iax	PROPN
ap-7678	575	21	,	,	PUNCT
ap-7678	575	22	a	a	DET
ap-7678	575	23	=	=	X
ap-7678	575	24	−2.10	−2.10	PROPN
ap-7678	575	25	77	77	NUM
ap-7678	575	26	carlos	carlos	PROPN
ap-7678	575	27	r.	r.	PROPN
ap-7678	575	28	handy	handy	PROPN
ap-7678	575	29	acta	acta	PROPN
ap-7678	575	30	polytechnica	polytechnica	PROPN
ap-7678	575	31	figure	figure	NOUN
ap-7678	575	32	11	11	NUM
ap-7678	575	33	.	.	PUNCT
ap-7678	576	1	log10(λ40(e	log10(λ40(e	NUM
ap-7678	576	2	)	)	PUNCT
ap-7678	576	3	)	)	PUNCT
ap-7678	577	1	for	for	ADP
ap-7678	577	2	pt	pt	NOUN
ap-7678	577	3	-	-	PUNCT
ap-7678	577	4	breaking	break	VERB
ap-7678	577	5	regime	regime	NOUN
ap-7678	577	6	,	,	PUNCT
ap-7678	577	7	−3.30	−3.30	VERB
ap-7678	577	8	≤	≤	NUM
ap-7678	577	9	a	a	DET
ap-7678	577	10	≤	≤	ADJ
ap-7678	577	11	−1.70	−1.70	PRON
ap-7678	577	12	for	for	ADP
ap-7678	577	13	v	v	NOUN
ap-7678	577	14	(	(	PUNCT
ap-7678	577	15	x	x	NOUN
ap-7678	577	16	)	)	PUNCT
ap-7678	577	17	=	=	SYM
ap-7678	577	18	ix3	ix3	PROPN
ap-7678	577	19	+	+	CCONJ
ap-7678	577	20	iax	iax	PROPN
ap-7678	577	21	,	,	PUNCT
ap-7678	577	22	a	a	DET
ap-7678	577	23	=	=	X
ap-7678	577	24	−1.70	−1.70	PART
ap-7678	577	25	figure	figure	NOUN
ap-7678	577	26	12	12	NUM
ap-7678	577	27	.	.	PUNCT
ap-7678	578	1	log10(λ40(e	log10(λ40(e	NUM
ap-7678	578	2	)	)	PUNCT
ap-7678	578	3	)	)	PUNCT
ap-7678	579	1	for	for	ADP
ap-7678	579	2	pt	pt	NOUN
ap-7678	579	3	-	-	PUNCT
ap-7678	579	4	breaking	break	VERB
ap-7678	579	5	regime	regime	NOUN
ap-7678	579	6	,	,	PUNCT
ap-7678	579	7	−3.30	−3.30	VERB
ap-7678	579	8	≤	≤	NUM
ap-7678	579	9	a	a	DET
ap-7678	579	10	≤	≤	ADJ
ap-7678	579	11	−1.70	−1.70	PRON
ap-7678	579	12	for	for	ADP
ap-7678	579	13	v	v	NOUN
ap-7678	579	14	(	(	PUNCT
ap-7678	579	15	x	x	NOUN
ap-7678	579	16	)	)	PUNCT
ap-7678	579	17	=	=	SYM
ap-7678	579	18	ix3	ix3	PROPN
ap-7678	579	19	+	+	CCONJ
ap-7678	579	20	iax	iax	PROPN
ap-7678	579	21	,	,	PUNCT
ap-7678	579	22	a	a	DET
ap-7678	579	23	=	=	NUM
ap-7678	579	24	0.00	0.00	NUM
ap-7678	579	25	−3.30	−3.30	NOUN
ap-7678	579	26	,	,	PUNCT
ap-7678	579	27	−2	−2	NOUN
ap-7678	579	28	,	,	PUNCT
ap-7678	579	29	90	90	NUM
ap-7678	579	30	,	,	PUNCT
ap-7678	579	31	−2.70	−2.70	NOUN
ap-7678	579	32	,	,	PUNCT
ap-7678	579	33	−2.50	−2.50	PROPN
ap-7678	579	34	,	,	PUNCT
ap-7678	579	35	−2.30	−2.30	ADV
ap-7678	579	36	,	,	PUNCT
ap-7678	579	37	−2.10	−2.10	PROPN
ap-7678	579	38	,	,	PUNCT
ap-7678	579	39	−1.70	−1.70	PRON
ap-7678	579	40	and	and	CCONJ
ap-7678	579	41	0	0	NUM
ap-7678	579	42	.	.	PUNCT
ap-7678	580	1	these	these	DET
ap-7678	580	2	parameter	parameter	NOUN
ap-7678	580	3	values	value	NOUN
ap-7678	580	4	were	be	AUX
ap-7678	580	5	chosen	choose	VERB
ap-7678	580	6	since	since	SCONJ
ap-7678	580	7	it	it	PRON
ap-7678	580	8	is	be	AUX
ap-7678	580	9	known	know	VERB
ap-7678	580	10	that	that	SCONJ
ap-7678	580	11	pt	pt	NOUN
ap-7678	580	12	-	-	PUNCT
ap-7678	580	13	symmetry	symmetry	NOUN
ap-7678	580	14	breaking	breaking	NOUN
ap-7678	580	15	occurs	occur	VERB
ap-7678	580	16	at	at	ADP
ap-7678	580	17	ac1	ac1	PROPN
ap-7678	580	18	=	=	PROPN
ap-7678	580	19	−2.611809356	−2.611809356	NOUN
ap-7678	580	20	(	(	PUNCT
ap-7678	580	21	as	as	ADV
ap-7678	580	22	well	well	ADV
ap-7678	580	23	as	as	ADP
ap-7678	580	24	at	at	ADP
ap-7678	580	25	ac2	ac2	PROPN
ap-7678	580	26	=	=	SYM
ap-7678	580	27	−5.375879629	−5.375879629	PROPN
ap-7678	580	28	)	)	PUNCT
ap-7678	581	1	[	[	X
ap-7678	581	2	8	8	NUM
ap-7678	581	3	,	,	PUNCT
ap-7678	581	4	25	25	NUM
ap-7678	581	5	]	]	PUNCT
ap-7678	581	6	.	.	PUNCT
ap-7678	582	1	figure	figure	NOUN
ap-7678	582	2	7	7	NUM
ap-7678	582	3	shows	show	VERB
ap-7678	582	4	the	the	DET
ap-7678	582	5	behavior	behavior	NOUN
ap-7678	582	6	of	of	ADP
ap-7678	582	7	the	the	DET
ap-7678	582	8	physical	physical	ADJ
ap-7678	582	9	energies	energy	NOUN
ap-7678	582	10	(	(	PUNCT
ap-7678	582	11	i.e.	i.e.	X
ap-7678	582	12	defined	define	VERB
ap-7678	582	13	by	by	ADP
ap-7678	582	14	the	the	DET
ap-7678	582	15	local	local	ADJ
ap-7678	582	16	minima	minima	PROPN
ap-7678	582	17	)	)	PUNCT
ap-7678	582	18	,	,	PUNCT
ap-7678	582	19	for	for	ADP
ap-7678	582	20	a	a	DET
ap-7678	582	21	=	=	SYM
ap-7678	582	22	−2.70	−2.70	PROPN
ap-7678	582	23	≈	≈	PROPN
ap-7678	582	24	ac1	ac1	PROPN
ap-7678	582	25	,	,	PUNCT
ap-7678	582	26	the	the	DET
ap-7678	582	27	critical	critical	ADJ
ap-7678	582	28	value	value	NOUN
ap-7678	582	29	for	for	ADP
ap-7678	582	30	the	the	DET
ap-7678	582	31	onset	onset	NOUN
ap-7678	582	32	of	of	ADP
ap-7678	582	33	symmetry	symmetry	NOUN
ap-7678	582	34	breaking	breaking	NOUN
ap-7678	582	35	.	.	PUNCT
ap-7678	583	1	for	for	ADP
ap-7678	583	2	this	this	DET
ap-7678	583	3	same	same	ADJ
ap-7678	583	4	a	a	DET
ap-7678	583	5	=	=	SYM
ap-7678	583	6	−2.70	−2.70	NOUN
ap-7678	583	7	,	,	PUNCT
ap-7678	583	8	figure	figure	NOUN
ap-7678	583	9	4	4	NUM
ap-7678	583	10	shows	show	VERB
ap-7678	583	11	the	the	DET
ap-7678	583	12	behavior	behavior	NOUN
ap-7678	583	13	of	of	ADP
ap-7678	583	14	successive	successive	ADJ
ap-7678	583	15	λn	λn	NOUN
ap-7678	583	16	(	(	PUNCT
ap-7678	583	17	e	e	NOUN
ap-7678	583	18	)	)	PUNCT
ap-7678	583	19	surfaces	surface	NOUN
ap-7678	583	20	for	for	ADP
ap-7678	583	21	n	n	NOUN
ap-7678	583	22	=	=	SYM
ap-7678	583	23	20	20	NUM
ap-7678	583	24	,	,	PUNCT
ap-7678	583	25	30	30	NUM
ap-7678	583	26	,	,	PUNCT
ap-7678	583	27	40	40	NUM
ap-7678	583	28	,	,	PUNCT
ap-7678	583	29	50	50	NUM
ap-7678	583	30	.	.	PUNCT
ap-7678	584	1	within	within	ADP
ap-7678	584	2	the	the	DET
ap-7678	584	3	two	two	NUM
ap-7678	584	4	dimensional	dimensional	ADJ
ap-7678	584	5	graphical	graphical	ADJ
ap-7678	584	6	renderings	rendering	NOUN
ap-7678	584	7	,	,	PUNCT
ap-7678	584	8	we	we	PRON
ap-7678	584	9	see	see	VERB
ap-7678	584	10	similar	similar	ADJ
ap-7678	584	11	behaviors	behavior	NOUN
ap-7678	584	12	to	to	ADP
ap-7678	584	13	that	that	PRON
ap-7678	584	14	in	in	ADP
ap-7678	584	15	figures	figure	NOUN
ap-7678	584	16	1	1	NUM
ap-7678	584	17	-	-	SYM
ap-7678	584	18	3	3	NUM
ap-7678	584	19	.	.	PUNCT
ap-7678	585	1	we	we	PRON
ap-7678	585	2	can	can	AUX
ap-7678	585	3	implement	implement	VERB
ap-7678	585	4	the	the	DET
ap-7678	585	5	same	same	ADJ
ap-7678	585	6	oppq	oppq	NOUN
ap-7678	585	7	-	-	PUNCT
ap-7678	585	8	bm	bm	NOUN
ap-7678	585	9	bounding	bounding	NOUN
ap-7678	585	10	analysis	analysis	NOUN
ap-7678	585	11	used	use	VERB
ap-7678	585	12	previously	previously	ADV
ap-7678	585	13	,	,	PUNCT
ap-7678	585	14	for	for	ADP
ap-7678	585	15	hermitian	hermitian	ADJ
ap-7678	585	16	systems	system	NOUN
ap-7678	585	17	,	,	PUNCT
ap-7678	585	18	to	to	PART
ap-7678	585	19	bound	bind	VERB
ap-7678	585	20	the	the	DET
ap-7678	585	21	real	real	ADJ
ap-7678	585	22	and	and	CCONJ
ap-7678	585	23	imaginary	imaginary	ADJ
ap-7678	585	24	parts	part	NOUN
ap-7678	585	25	of	of	ADP
ap-7678	585	26	the	the	DET
ap-7678	585	27	discrete	discrete	ADJ
ap-7678	585	28	state	state	NOUN
ap-7678	585	29	real	real	ADJ
ap-7678	585	30	/	/	SYM
ap-7678	585	31	complex	complex	ADJ
ap-7678	585	32	energies	energy	NOUN
ap-7678	585	33	.	.	PUNCT
ap-7678	586	1	the	the	DET
ap-7678	586	2	nesting	nesting	NOUN
ap-7678	586	3	of	of	ADP
ap-7678	586	4	the	the	DET
ap-7678	586	5	respecive	respecive	ADJ
ap-7678	586	6	λn	λn	NOUN
ap-7678	586	7	(	(	PUNCT
ap-7678	586	8	er	er	INTJ
ap-7678	586	9	,	,	PUNCT
ap-7678	586	10	ei	ei	NOUN
ap-7678	586	11	)	)	PUNCT
ap-7678	586	12	in	in	ADP
ap-7678	586	13	figure	figure	NOUN
ap-7678	586	14	4	4	NUM
ap-7678	586	15	shows	show	VERB
ap-7678	586	16	the	the	DET
ap-7678	586	17	viability	viability	NOUN
ap-7678	586	18	of	of	ADP
ap-7678	586	19	the	the	DET
ap-7678	586	20	previous	previous	ADJ
ap-7678	586	21	oppq	oppq	NOUN
ap-7678	586	22	-	-	PUNCT
ap-7678	586	23	bm	bm	NOUN
ap-7678	586	24	analysis	analysis	NOUN
ap-7678	586	25	for	for	ADP
ap-7678	586	26	bounding	bound	VERB
ap-7678	586	27	the	the	DET
ap-7678	586	28	real	real	ADJ
ap-7678	586	29	and	and	CCONJ
ap-7678	586	30	imaginary	imaginary	ADJ
ap-7678	586	31	parts	part	NOUN
ap-7678	586	32	of	of	ADP
ap-7678	586	33	the	the	DET
ap-7678	586	34	complex	complex	ADJ
ap-7678	586	35	-	-	PUNCT
ap-7678	586	36	plane	plane	NOUN
ap-7678	586	37	energies	energy	NOUN
ap-7678	586	38	.	.	PUNCT
ap-7678	587	1	this	this	PRON
ap-7678	587	2	is	be	AUX
ap-7678	587	3	the	the	DET
ap-7678	587	4	focus	focus	NOUN
ap-7678	587	5	of	of	ADP
ap-7678	587	6	a	a	DET
ap-7678	587	7	future	future	ADJ
ap-7678	587	8	work	work	NOUN
ap-7678	587	9	.	.	PUNCT
ap-7678	588	1	we	we	PRON
ap-7678	588	2	note	note	VERB
ap-7678	588	3	that	that	SCONJ
ap-7678	588	4	for	for	ADP
ap-7678	588	5	a	a	DET
ap-7678	588	6	=	=	SYM
ap-7678	588	7	0	0	NUM
ap-7678	588	8	,	,	PUNCT
ap-7678	588	9	the	the	DET
ap-7678	588	10	local	local	ADJ
ap-7678	588	11	minima	minima	NOUN
ap-7678	588	12	in	in	ADP
ap-7678	588	13	figure	figure	NOUN
ap-7678	588	14	12	12	NUM
ap-7678	588	15	correspond	correspond	NOUN
ap-7678	588	16	,	,	PUNCT
ap-7678	588	17	approximately	approximately	ADV
ap-7678	588	18	(	(	PUNCT
ap-7678	588	19	due	due	ADP
ap-7678	588	20	to	to	ADP
ap-7678	588	21	the	the	DET
ap-7678	588	22	low	low	ADJ
ap-7678	588	23	order	order	NOUN
ap-7678	588	24	,	,	PUNCT
ap-7678	588	25	n	n	NOUN
ap-7678	588	26	=	=	NUM
ap-7678	588	27	40	40	NUM
ap-7678	588	28	)	)	PUNCT
ap-7678	588	29	to	to	ADP
ap-7678	588	30	the	the	DET
ap-7678	588	31	pt	pt	ADJ
ap-7678	588	32	-	-	ADJ
ap-7678	588	33	symmetric	symmetric	ADJ
ap-7678	588	34	states	state	NOUN
ap-7678	588	35	with	with	ADP
ap-7678	588	36	energies	energy	NOUN
ap-7678	588	37	e0	e0	PROPN
ap-7678	588	38	=	=	SYM
ap-7678	588	39	1.15626707198811	1.15626707198811	NUM
ap-7678	588	40	,	,	PUNCT
ap-7678	588	41	e2	e2	PROPN
ap-7678	588	42	=	=	SYM
ap-7678	588	43	4.10922875280956	4.10922875280956	NUM
ap-7678	588	44	,	,	PUNCT
ap-7678	588	45	e3	e3	NOUN
ap-7678	588	46	=	=	SYM
ap-7678	588	47	7.56227385497590	7.56227385497590	NUM
ap-7678	588	48	and	and	CCONJ
ap-7678	588	49	e4	e4	PROPN
ap-7678	588	50	=	=	SYM
ap-7678	588	51	11.31442182025857	11.31442182025857	NUM
ap-7678	588	52	(	(	PUNCT
ap-7678	588	53	not	not	PART
ap-7678	588	54	shown	show	VERB
ap-7678	588	55	)	)	PUNCT
ap-7678	588	56	.	.	PUNCT
ap-7678	589	1	these	these	PRON
ap-7678	589	2	were	be	AUX
ap-7678	589	3	determined	determine	VERB
ap-7678	589	4	by	by	ADP
ap-7678	589	5	emm	emm	PROPN
ap-7678	590	1	[	[	X
ap-7678	590	2	8	8	NUM
ap-7678	590	3	]	]	PUNCT
ap-7678	590	4	and	and	CCONJ
ap-7678	590	5	oppq	oppq	NOUN
ap-7678	590	6	-	-	PUNCT
ap-7678	590	7	am	be	AUX
ap-7678	590	8	[	[	X
ap-7678	590	9	10	10	NUM
ap-7678	590	10	]	]	PUNCT
ap-7678	590	11	.	.	PUNCT
ap-7678	591	1	11	11	NUM
ap-7678	591	2	.	.	PUNCT
ap-7678	592	1	conclusions	conclusion	NOUN
ap-7678	592	2	we	we	PRON
ap-7678	592	3	have	have	AUX
ap-7678	592	4	demonstrated	demonstrate	VERB
ap-7678	592	5	the	the	DET
ap-7678	592	6	effectiveness	effectiveness	NOUN
ap-7678	592	7	of	of	ADP
ap-7678	592	8	a	a	DET
ap-7678	592	9	new	new	ADJ
ap-7678	592	10	eigenenergy	eigenenergy	NOUN
ap-7678	592	11	bounding	bounding	NOUN
ap-7678	592	12	procedure	procedure	NOUN
ap-7678	592	13	implementable	implementable	ADJ
ap-7678	592	14	for	for	ADP
ap-7678	592	15	multidimensional	multidimensional	ADJ
ap-7678	592	16	discrete	discrete	ADJ
ap-7678	592	17	states	state	NOUN
ap-7678	592	18	regardless	regardless	ADV
ap-7678	592	19	of	of	ADP
ap-7678	592	20	the	the	DET
ap-7678	592	21	hermitian	hermitian	ADJ
ap-7678	592	22	or	or	CCONJ
ap-7678	592	23	non	non	ADJ
ap-7678	592	24	-	-	ADJ
ap-7678	592	25	hermitian	hermitian	ADJ
ap-7678	592	26	character	character	NOUN
ap-7678	592	27	of	of	ADP
ap-7678	592	28	the	the	DET
ap-7678	592	29	associated	associated	ADJ
ap-7678	592	30	schrödinger	schrödinger	ADJ
ap-7678	592	31	operator	operator	NOUN
ap-7678	592	32	.	.	PUNCT
ap-7678	593	1	the	the	DET
ap-7678	593	2	discussion	discussion	NOUN
ap-7678	593	3	centered	center	VERB
ap-7678	593	4	on	on	ADP
ap-7678	593	5	several	several	ADJ
ap-7678	593	6	one	one	NUM
ap-7678	593	7	dimensional	dimensional	ADJ
ap-7678	593	8	systems	system	NOUN
ap-7678	593	9	of	of	ADP
ap-7678	593	10	this	this	DET
ap-7678	593	11	type	type	NOUN
ap-7678	593	12	.	.	PUNCT
ap-7678	594	1	the	the	DET
ap-7678	594	2	extension	extension	NOUN
ap-7678	594	3	to	to	ADP
ap-7678	594	4	multidimensions	multidimension	NOUN
ap-7678	594	5	have	have	AUX
ap-7678	594	6	been	be	AUX
ap-7678	594	7	given	give	VERB
ap-7678	594	8	elsewhere	elsewhere	ADV
ap-7678	594	9	[	[	X
ap-7678	594	10	9	9	NUM
ap-7678	594	11	]	]	PUNCT
ap-7678	594	12	.	.	PUNCT
ap-7678	595	1	the	the	DET
ap-7678	595	2	approach	approach	NOUN
ap-7678	595	3	advocated	advocate	VERB
ap-7678	595	4	here	here	ADV
ap-7678	595	5	is	be	AUX
ap-7678	595	6	purely	purely	ADV
ap-7678	595	7	algebraic	algebraic	ADJ
ap-7678	595	8	.	.	PUNCT
ap-7678	596	1	acknowledgements	acknowledgement	VERB
ap-7678	596	2	the	the	DET
ap-7678	596	3	author	author	NOUN
ap-7678	596	4	is	be	AUX
ap-7678	596	5	greatful	greatful	ADJ
ap-7678	596	6	to	to	ADP
ap-7678	596	7	the	the	DET
ap-7678	596	8	organizers	organizer	NOUN
ap-7678	596	9	of	of	ADP
ap-7678	596	10	the	the	DET
ap-7678	596	11	2021	2021	NUM
ap-7678	596	12	conference	conference	NOUN
ap-7678	596	13	on	on	ADP
ap-7678	596	14	analytic	analytic	ADJ
ap-7678	596	15	and	and	CCONJ
ap-7678	596	16	algebraic	algebraic	ADJ
ap-7678	596	17	methods	method	NOUN
ap-7678	596	18	in	in	ADP
ap-7678	596	19	physics	physics	PROPN
ap-7678	596	20	xviii	xviii	PROPN
ap-7678	596	21	,	,	PUNCT
ap-7678	596	22	particularly	particularly	ADV
ap-7678	596	23	dr	dr	PROPN
ap-7678	596	24	.	.	PROPN
ap-7678	596	25	miloslav	miloslav	PROPN
ap-7678	596	26	znojil	znojil	PROPN
ap-7678	596	27	and	and	CCONJ
ap-7678	596	28	dr	dr	PROPN
ap-7678	596	29	.	.	PROPN
ap-7678	596	30	vit	vit	PROPN
ap-7678	596	31	jakubsky	jakubsky	PROPN
ap-7678	596	32	,	,	PUNCT
ap-7678	596	33	and	and	CCONJ
ap-7678	596	34	the	the	DET
ap-7678	596	35	support	support	NOUN
ap-7678	596	36	from	from	ADP
ap-7678	596	37	the	the	DET
ap-7678	596	38	doppler	doppler	NOUN
ap-7678	596	39	institute	institute	NOUN
ap-7678	596	40	of	of	ADP
ap-7678	596	41	the	the	DET
ap-7678	596	42	czech	czech	PROPN
ap-7678	596	43	academy	academy	PROPN
ap-7678	596	44	of	of	ADP
ap-7678	596	45	sciences	sciences	PROPN
ap-7678	596	46	.	.	PUNCT
ap-7678	597	1	additional	additional	ADJ
ap-7678	597	2	remarks	remark	NOUN
ap-7678	597	3	from	from	ADP
ap-7678	597	4	dr	dr	PROPN
ap-7678	597	5	.	.	PROPN
ap-7678	597	6	daniel	daniel	PROPN
ap-7678	597	7	bessis	bessis	PROPN
ap-7678	597	8	,	,	PUNCT
ap-7678	597	9	dr	dr	PROPN
ap-7678	597	10	.	.	PROPN
ap-7678	597	11	john	john	PROPN
ap-7678	597	12	r.	r.	PROPN
ap-7678	597	13	klauder	klauder	PROPN
ap-7678	597	14	,	,	PUNCT
ap-7678	597	15	dr	dr	PROPN
ap-7678	597	16	.	.	PROPN
ap-7678	597	17	jerzy	jerzy	PROPN
ap-7678	597	18	cioslowski	cioslowski	PROPN
ap-7678	597	19	,	,	PUNCT
ap-7678	597	20	and	and	CCONJ
ap-7678	597	21	dr	dr	PROPN
ap-7678	597	22	.	.	PROPN
ap-7678	597	23	a.	a.	PROPN
ap-7678	597	24	turbiner	turbiner	NOUN
ap-7678	597	25	are	be	AUX
ap-7678	597	26	greatly	greatly	ADV
ap-7678	597	27	appreciated	appreciate	VERB
ap-7678	597	28	.	.	PUNCT
ap-7678	598	1	technical	technical	ADJ
ap-7678	598	2	support	support	NOUN
ap-7678	598	3	from	from	ADP
ap-7678	598	4	dr	dr	PROPN
ap-7678	598	5	.	.	PROPN
ap-7678	598	6	maribel	maribel	PROPN
ap-7678	598	7	handy	handy	PROPN
ap-7678	598	8	is	be	AUX
ap-7678	598	9	also	also	ADV
ap-7678	598	10	greatly	greatly	ADV
ap-7678	598	11	appreciated	appreciate	VERB
ap-7678	598	12	.	.	PUNCT
ap-7678	599	1	references	reference	NOUN
ap-7678	599	2	[	[	X
ap-7678	599	3	1	1	NUM
ap-7678	599	4	]	]	X
ap-7678	599	5	w.	w.	PROPN
ap-7678	599	6	ritz	ritz	PROPN
ap-7678	599	7	.	.	PROPN
ap-7678	599	8	über	über	PROPN
ap-7678	599	9	eine	eine	PROPN
ap-7678	599	10	neue	neue	PROPN
ap-7678	599	11	methode	methode	PROPN
ap-7678	599	12	zur	zur	PROPN
ap-7678	599	13	lösung	lösung	PROPN
ap-7678	599	14	gewisser	gewisser	PROPN
ap-7678	599	15	variationsprobleme	variationsprobleme	PROPN
ap-7678	599	16	der	der	PROPN
ap-7678	599	17	mathematischen	mathematischen	PROPN
ap-7678	599	18	physik	physik	PROPN
ap-7678	599	19	.	.	PROPN
ap-7678	600	1	journal	journal	PROPN
ap-7678	600	2	für	für	PROPN
ap-7678	600	3	die	die	VERB
ap-7678	600	4	reine	reine	PROPN
ap-7678	600	5	und	und	PROPN
ap-7678	600	6	angewandte	angewandte	PROPN
ap-7678	600	7	mathematik	mathematik	PROPN
ap-7678	600	8	1909(135):1–61	1909(135):1–61	PROPN
ap-7678	600	9	,	,	PUNCT
ap-7678	600	10	1909	1909	NUM
ap-7678	600	11	.	.	PUNCT
ap-7678	601	1	https://doi.org/10.1515/crll.1909.135.1	https://doi.org/10.1515/crll.1909.135.1	PROPN
ap-7678	601	2	.	.	PUNCT
ap-7678	602	1	[	[	X
ap-7678	602	2	2	2	NUM
ap-7678	602	3	]	]	X
ap-7678	602	4	g.	g.	NOUN
ap-7678	602	5	temple	temple	PROPN
ap-7678	602	6	.	.	PUNCT
ap-7678	603	1	the	the	DET
ap-7678	603	2	theory	theory	NOUN
ap-7678	603	3	of	of	ADP
ap-7678	603	4	rayleigh	rayleigh	PROPN
ap-7678	603	5	’s	’s	PART
ap-7678	603	6	principle	principle	NOUN
ap-7678	603	7	as	as	SCONJ
ap-7678	603	8	applied	apply	VERB
ap-7678	603	9	to	to	ADP
ap-7678	603	10	continuous	continuous	ADJ
ap-7678	603	11	systems	system	NOUN
ap-7678	603	12	.	.	PUNCT
ap-7678	604	1	proceedings	proceeding	NOUN
ap-7678	604	2	of	of	ADP
ap-7678	604	3	the	the	DET
ap-7678	604	4	royal	royal	ADJ
ap-7678	604	5	society	society	NOUN
ap-7678	604	6	of	of	ADP
ap-7678	604	7	london	london	PROPN
ap-7678	604	8	series	series	PROPN
ap-7678	604	9	a	a	PROPN
ap-7678	604	10	,	,	PUNCT
ap-7678	604	11	containing	contain	VERB
ap-7678	604	12	papers	paper	NOUN
ap-7678	604	13	of	of	ADP
ap-7678	604	14	a	a	DET
ap-7678	604	15	mathematical	mathematical	ADJ
ap-7678	604	16	and	and	CCONJ
ap-7678	604	17	physical	physical	ADJ
ap-7678	604	18	character	character	NOUN
ap-7678	604	19	119(782):276–293	119(782):276–293	NUM
ap-7678	604	20	,	,	PUNCT
ap-7678	604	21	1928	1928	NUM
ap-7678	604	22	.	.	PUNCT
ap-7678	605	1	http://www.jstor.org/stable/95047	http://www.jstor.org/stable/95047	NOUN
ap-7678	605	2	.	.	PUNCT
ap-7678	606	1	[	[	X
ap-7678	606	2	3	3	X
ap-7678	606	3	]	]	PUNCT
ap-7678	606	4	m.	m.	NOUN
ap-7678	606	5	g.	g.	PROPN
ap-7678	606	6	marmorino	marmorino	PROPN
ap-7678	606	7	,	,	PUNCT
ap-7678	606	8	a.	a.	NOUN
ap-7678	606	9	almayouf	almayouf	PROPN
ap-7678	606	10	,	,	PUNCT
ap-7678	606	11	t.	t.	PROPN
ap-7678	606	12	krause	krause	PROPN
ap-7678	606	13	,	,	PUNCT
ap-7678	606	14	d.	d.	PROPN
ap-7678	606	15	le	le	PROPN
ap-7678	606	16	.	.	PUNCT
ap-7678	607	1	optimization	optimization	NOUN
ap-7678	607	2	of	of	ADP
ap-7678	607	3	the	the	DET
ap-7678	607	4	temple	temple	NOUN
ap-7678	607	5	lower	lower	ADV
ap-7678	607	6	bound	bind	VERB
ap-7678	607	7	.	.	PUNCT
ap-7678	608	1	journal	journal	PROPN
ap-7678	608	2	of	of	ADP
ap-7678	608	3	mathematical	mathematical	ADJ
ap-7678	608	4	chemistry	chemistry	NOUN
ap-7678	608	5	50(4):833–842	50(4):833–842	PROPN
ap-7678	608	6	,	,	PUNCT
ap-7678	608	7	2012	2012	NUM
ap-7678	608	8	.	.	PUNCT
ap-7678	609	1	https://doi.org/10.1007/s10910-011-9927-z	https://doi.org/10.1007/s10910-011-9927-z	NOUN
ap-7678	609	2	.	.	PUNCT
ap-7678	610	1	[	[	X
ap-7678	610	2	4	4	X
ap-7678	610	3	]	]	X
ap-7678	610	4	r.	r.	PROPN
ap-7678	610	5	martinazzo	martinazzo	PROPN
ap-7678	610	6	,	,	PUNCT
ap-7678	610	7	e.	e.	PROPN
ap-7678	610	8	pollak	pollak	PROPN
ap-7678	610	9	.	.	PUNCT
ap-7678	611	1	lower	low	ADJ
ap-7678	611	2	bounds	bound	NOUN
ap-7678	611	3	to	to	ADP
ap-7678	611	4	eigenvalues	eigenvalue	NOUN
ap-7678	611	5	of	of	ADP
ap-7678	611	6	the	the	DET
ap-7678	611	7	schrödinger	schrödinger	ADJ
ap-7678	611	8	equation	equation	NOUN
ap-7678	611	9	by	by	ADP
ap-7678	611	10	solution	solution	NOUN
ap-7678	611	11	of	of	ADP
ap-7678	611	12	a	a	DET
ap-7678	611	13	90	90	NUM
ap-7678	611	14	-	-	PUNCT
ap-7678	611	15	y	y	NOUN
ap-7678	611	16	challenge	challenge	NOUN
ap-7678	611	17	.	.	PUNCT
ap-7678	612	1	proceedings	proceeding	NOUN
ap-7678	612	2	of	of	ADP
ap-7678	612	3	the	the	DET
ap-7678	612	4	national	national	PROPN
ap-7678	612	5	academy	academy	PROPN
ap-7678	612	6	of	of	ADP
ap-7678	612	7	sciences	science	NOUN
ap-7678	612	8	117(28):16181–16186	117(28):16181–16186	NUM
ap-7678	612	9	,	,	PUNCT
ap-7678	612	10	2020	2020	NUM
ap-7678	612	11	.	.	PUNCT
ap-7678	613	1	https://doi.org/10.1073/pnas.2007093117	https://doi.org/10.1073/pnas.2007093117	X
ap-7678	613	2	.	.	PUNCT
ap-7678	614	1	[	[	X
ap-7678	614	2	5	5	NUM
ap-7678	614	3	]	]	PUNCT
ap-7678	614	4	c.	c.	PROPN
ap-7678	614	5	m.	m.	PROPN
ap-7678	614	6	bender	bender	PROPN
ap-7678	614	7	,	,	PUNCT
ap-7678	614	8	s.	s.	PROPN
ap-7678	614	9	boettcher	boettcher	PROPN
ap-7678	614	10	.	.	PUNCT
ap-7678	615	1	real	real	ADJ
ap-7678	615	2	spectra	spectra	NOUN
ap-7678	615	3	in	in	ADP
ap-7678	615	4	non	non	ADJ
ap-7678	615	5	-	-	ADJ
ap-7678	615	6	hermitian	hermitian	ADJ
ap-7678	615	7	hamiltonians	hamiltonian	NOUN
ap-7678	615	8	having	have	VERB
ap-7678	615	9	pt	pt	PRON
ap-7678	615	10	symmetry	symmetry	NOUN
ap-7678	615	11	.	.	PUNCT
ap-7678	616	1	physical	physical	ADJ
ap-7678	616	2	review	review	NOUN
ap-7678	616	3	letters	letter	NOUN
ap-7678	616	4	80:5243–5246	80:5243–5246	NUM
ap-7678	616	5	,	,	PUNCT
ap-7678	616	6	1998	1998	NUM
ap-7678	616	7	.	.	PUNCT
ap-7678	617	1	https://doi.org/10.1103/physrevlett.80.5243	https://doi.org/10.1103/physrevlett.80.5243	NOUN
ap-7678	617	2	.	.	PUNCT
ap-7678	618	1	[	[	X
ap-7678	618	2	6	6	NUM
ap-7678	618	3	]	]	PUNCT
ap-7678	618	4	p.	p.	NOUN
ap-7678	618	5	dorey	dorey	PROPN
ap-7678	618	6	,	,	PUNCT
ap-7678	618	7	c.	c.	PROPN
ap-7678	618	8	dunning	dunning	NOUN
ap-7678	618	9	,	,	PUNCT
ap-7678	618	10	r.	r.	NOUN
ap-7678	618	11	tateo	tateo	NOUN
ap-7678	618	12	.	.	PUNCT
ap-7678	619	1	supersymmetry	supersymmetry	NOUN
ap-7678	619	2	and	and	CCONJ
ap-7678	619	3	the	the	DET
ap-7678	619	4	spontaneous	spontaneous	ADJ
ap-7678	619	5	breakdown	breakdown	NOUN
ap-7678	619	6	of	of	ADP
ap-7678	619	7	pt	pt	NOUN
ap-7678	619	8	symmetry	symmetry	NOUN
ap-7678	619	9	.	.	PUNCT
ap-7678	620	1	journal	journal	PROPN
ap-7678	620	2	of	of	ADP
ap-7678	620	3	physics	physics	PROPN
ap-7678	620	4	a	a	PRON
ap-7678	620	5	:	:	PUNCT
ap-7678	620	6	mathematical	mathematical	ADJ
ap-7678	620	7	and	and	CCONJ
ap-7678	620	8	general	general	ADJ
ap-7678	620	9	34(28):l391	34(28):l391	NUM
ap-7678	620	10	–	–	PUNCT
ap-7678	620	11	l400	l400	PROPN
ap-7678	620	12	,	,	PUNCT
ap-7678	620	13	2001	2001	NUM
ap-7678	620	14	.	.	PUNCT
ap-7678	621	1	https://doi.org/10.1088/0305-4470/34/28/102	https://doi.org/10.1088/0305-4470/34/28/102	PROPN
ap-7678	621	2	.	.	PUNCT
ap-7678	622	1	78	78	NUM
ap-7678	623	1	https://doi.org/10.1515/crll.1909.135.1	https://doi.org/10.1515/crll.1909.135.1	PROPN
ap-7678	623	2	http://www.jstor.org/stable/95047	http://www.jstor.org/stable/95047	NOUN
ap-7678	624	1	https://doi.org/10.1007/s10910-011-9927-z	https://doi.org/10.1007/s10910-011-9927-z	PROPN
ap-7678	625	1	https://doi.org/10.1073/pnas.2007093117	https://doi.org/10.1073/pnas.2007093117	PROPN
ap-7678	625	2	https://doi.org/10.1103/physrevlett.80.5243	https://doi.org/10.1103/physrevlett.80.5243	VERB
ap-7678	625	3	https://doi.org/10.1088/0305-4470/34/28/102	https://doi.org/10.1088/0305-4470/34/28/102	PROPN
ap-7678	625	4	vol	vol	NOUN
ap-7678	625	5	.	.	PUNCT
ap-7678	626	1	62	62	NUM
ap-7678	626	2	no	no	INTJ
ap-7678	626	3	.	.	PUNCT
ap-7678	627	1	1/2022	1/2022	NUM
ap-7678	627	2	algebraic	algebraic	ADJ
ap-7678	627	3	eigenenergy	eigenenergy	NOUN
ap-7678	627	4	bounding	bounding	NOUN
ap-7678	627	5	method	method	NOUN
ap-7678	627	6	[	[	X
ap-7678	627	7	7	7	X
ap-7678	627	8	]	]	X
ap-7678	627	9	c.	c.	PROPN
ap-7678	627	10	m.	m.	PROPN
ap-7678	627	11	bender	bender	PROPN
ap-7678	627	12	.	.	PUNCT
ap-7678	628	1	making	make	VERB
ap-7678	628	2	sense	sense	NOUN
ap-7678	628	3	of	of	ADP
ap-7678	628	4	non	non	ADJ
ap-7678	628	5	-	-	ADJ
ap-7678	628	6	hermitian	hermitian	ADJ
ap-7678	628	7	hamiltonians	hamiltonian	NOUN
ap-7678	628	8	.	.	PUNCT
ap-7678	629	1	reports	report	NOUN
ap-7678	629	2	on	on	ADP
ap-7678	629	3	progress	progress	NOUN
ap-7678	629	4	in	in	ADP
ap-7678	629	5	physics	physics	NOUN
ap-7678	629	6	70(6):947–1018	70(6):947–1018	PROPN
ap-7678	629	7	,	,	PUNCT
ap-7678	629	8	2007	2007	NUM
ap-7678	629	9	.	.	PUNCT
ap-7678	630	1	https://doi.org/10.1088/0034-4885/70/6/r03	https://doi.org/10.1088/0034-4885/70/6/r03	PROPN
ap-7678	630	2	.	.	PUNCT
ap-7678	631	1	[	[	X
ap-7678	631	2	8	8	NUM
ap-7678	631	3	]	]	X
ap-7678	631	4	c.	c.	PROPN
ap-7678	631	5	r.	r.	PROPN
ap-7678	631	6	handy	handy	PROPN
ap-7678	631	7	.	.	PUNCT
ap-7678	632	1	generating	generate	VERB
ap-7678	632	2	converging	converge	VERB
ap-7678	632	3	bounds	bound	NOUN
ap-7678	632	4	to	to	ADP
ap-7678	632	5	the	the	DET
ap-7678	632	6	(	(	PUNCT
ap-7678	632	7	complex	complex	ADJ
ap-7678	632	8	)	)	PUNCT
ap-7678	632	9	discrete	discrete	ADJ
ap-7678	632	10	states	state	NOUN
ap-7678	632	11	of	of	ADP
ap-7678	632	12	p	p	NOUN
ap-7678	632	13	2+ix3+iαx	2+ix3+iαx	NUM
ap-7678	632	14	hamiltonian	hamiltonian	NOUN
ap-7678	632	15	.	.	PUNCT
ap-7678	633	1	journal	journal	PROPN
ap-7678	633	2	of	of	ADP
ap-7678	633	3	physics	physics	PROPN
ap-7678	633	4	a	a	PRON
ap-7678	633	5	:	:	PUNCT
ap-7678	633	6	mathematical	mathematical	ADJ
ap-7678	633	7	and	and	CCONJ
ap-7678	633	8	general	general	ADJ
ap-7678	633	9	34(24):5065–5081	34(24):5065–5081	NUM
ap-7678	633	10	,	,	PUNCT
ap-7678	633	11	2001	2001	NUM
ap-7678	633	12	.	.	PUNCT
ap-7678	634	1	https://doi.org/10.1088/0305-4470/34/24/305	https://doi.org/10.1088/0305-4470/34/24/305	NOUN
ap-7678	634	2	.	.	PUNCT
ap-7678	635	1	[	[	X
ap-7678	635	2	9	9	NUM
ap-7678	635	3	]	]	X
ap-7678	635	4	c.	c.	PROPN
ap-7678	635	5	r.	r.	PROPN
ap-7678	635	6	handy	handy	PROPN
ap-7678	635	7	.	.	PUNCT
ap-7678	636	1	exact	exact	ADJ
ap-7678	636	2	christoffel	christoffel	NOUN
ap-7678	636	3	-	-	PUNCT
ap-7678	636	4	darboux	darboux	VERB
ap-7678	636	5	expansions	expansion	NOUN
ap-7678	636	6	:	:	PUNCT
ap-7678	636	7	a	a	DET
ap-7678	636	8	new	new	ADJ
ap-7678	636	9	,	,	PUNCT
ap-7678	636	10	multidimensional	multidimensional	ADJ
ap-7678	636	11	,	,	PUNCT
ap-7678	636	12	algebraic	algebraic	ADJ
ap-7678	636	13	,	,	PUNCT
ap-7678	636	14	eigenenergy	eigenenergy	NOUN
ap-7678	636	15	bounding	bounding	NOUN
ap-7678	636	16	method	method	NOUN
ap-7678	636	17	.	.	PUNCT
ap-7678	637	1	physica	physica	NOUN
ap-7678	637	2	scripta	scripta	PROPN
ap-7678	637	3	96(7):075201	96(7):075201	NUM
ap-7678	637	4	,	,	PUNCT
ap-7678	637	5	2021	2021	NUM
ap-7678	637	6	.	.	PUNCT
ap-7678	638	1	https://doi.org/10.1088/1402-4896/abf67e	https://doi.org/10.1088/1402-4896/abf67e	NOUN
ap-7678	638	2	.	.	PUNCT
ap-7678	639	1	[	[	X
ap-7678	639	2	10	10	NUM
ap-7678	639	3	]	]	X
ap-7678	639	4	c.	c.	PROPN
ap-7678	639	5	r.	r.	PROPN
ap-7678	639	6	handy	handy	PROPN
ap-7678	639	7	,	,	PUNCT
ap-7678	639	8	d.	d.	PROPN
ap-7678	639	9	vrinceanu	vrinceanu	PROPN
ap-7678	639	10	.	.	PUNCT
ap-7678	640	1	orthogonal	orthogonal	ADJ
ap-7678	640	2	polynomial	polynomial	ADJ
ap-7678	640	3	projection	projection	NOUN
ap-7678	640	4	quantization	quantization	NOUN
ap-7678	640	5	:	:	PUNCT
ap-7678	640	6	a	a	DET
ap-7678	640	7	new	new	ADJ
ap-7678	640	8	hill	hill	NOUN
ap-7678	640	9	determinant	determinant	ADJ
ap-7678	640	10	method	method	NOUN
ap-7678	640	11	.	.	PUNCT
ap-7678	641	1	journal	journal	PROPN
ap-7678	641	2	of	of	ADP
ap-7678	641	3	physics	physics	PROPN
ap-7678	641	4	a	a	PRON
ap-7678	641	5	:	:	PUNCT
ap-7678	641	6	mathematical	mathematical	ADJ
ap-7678	641	7	and	and	CCONJ
ap-7678	641	8	theoretical	theoretical	ADJ
ap-7678	641	9	46(13):135202	46(13):135202	PROPN
ap-7678	641	10	,	,	PUNCT
ap-7678	641	11	2013	2013	NUM
ap-7678	641	12	.	.	PUNCT
ap-7678	642	1	https://doi.org/10.1088/1751-8113/46/13/135202	https://doi.org/10.1088/1751-8113/46/13/135202	NOUN
ap-7678	642	2	.	.	PUNCT
ap-7678	643	1	[	[	X
ap-7678	643	2	11	11	NUM
ap-7678	643	3	]	]	X
ap-7678	643	4	c.	c.	PROPN
ap-7678	643	5	r.	r.	PROPN
ap-7678	643	6	handy	handy	PROPN
ap-7678	643	7	,	,	PUNCT
ap-7678	643	8	d.	d.	PROPN
ap-7678	643	9	vrinceanu	vrinceanu	PROPN
ap-7678	643	10	.	.	PUNCT
ap-7678	644	1	rapidly	rapidly	ADV
ap-7678	644	2	converging	converge	VERB
ap-7678	644	3	bound	bind	VERB
ap-7678	644	4	state	state	NOUN
ap-7678	644	5	eigenenergies	eigenenergie	NOUN
ap-7678	644	6	for	for	ADP
ap-7678	644	7	the	the	DET
ap-7678	644	8	two	two	NUM
ap-7678	644	9	dimensional	dimensional	ADJ
ap-7678	644	10	quantum	quantum	ADJ
ap-7678	644	11	dipole	dipole	NOUN
ap-7678	644	12	.	.	PUNCT
ap-7678	645	1	journal	journal	PROPN
ap-7678	645	2	of	of	ADP
ap-7678	645	3	physics	physics	PROPN
ap-7678	645	4	b	b	PROPN
ap-7678	645	5	:	:	PUNCT
ap-7678	645	6	atomic	atomic	ADJ
ap-7678	645	7	,	,	PUNCT
ap-7678	645	8	molecular	molecular	ADJ
ap-7678	645	9	and	and	CCONJ
ap-7678	645	10	optical	optical	ADJ
ap-7678	645	11	physics	physics	NOUN
ap-7678	645	12	46(11):115002	46(11):115002	ADP
ap-7678	645	13	,	,	PUNCT
ap-7678	645	14	2013	2013	NUM
ap-7678	645	15	.	.	PUNCT
ap-7678	646	1	https://doi.org/10.1088/0953-4075/46/11/115002	https://doi.org/10.1088/0953-4075/46/11/115002	PROPN
ap-7678	646	2	.	.	PUNCT
ap-7678	647	1	[	[	X
ap-7678	647	2	12	12	NUM
ap-7678	647	3	]	]	X
ap-7678	647	4	j.	j.	PROPN
ap-7678	647	5	le	le	PROPN
ap-7678	647	6	guillou	guillou	PROPN
ap-7678	647	7	,	,	PUNCT
ap-7678	647	8	j.	j.	PROPN
ap-7678	647	9	zinn	zinn	PROPN
ap-7678	647	10	-	-	PUNCT
ap-7678	647	11	justin	justin	PROPN
ap-7678	647	12	.	.	PUNCT
ap-7678	648	1	the	the	DET
ap-7678	648	2	hydrogen	hydrogen	NOUN
ap-7678	648	3	atom	atom	NOUN
ap-7678	648	4	in	in	ADP
ap-7678	648	5	strong	strong	ADJ
ap-7678	648	6	magnetic	magnetic	ADJ
ap-7678	648	7	fields	field	NOUN
ap-7678	648	8	:	:	PUNCT
ap-7678	648	9	summation	summation	NOUN
ap-7678	648	10	of	of	ADP
ap-7678	648	11	the	the	DET
ap-7678	648	12	weak	weak	ADJ
ap-7678	648	13	field	field	NOUN
ap-7678	648	14	series	series	NOUN
ap-7678	648	15	expansion	expansion	NOUN
ap-7678	648	16	.	.	PUNCT
ap-7678	649	1	annals	annal	NOUN
ap-7678	649	2	of	of	ADP
ap-7678	649	3	physics	physics	NOUN
ap-7678	649	4	147(1):57–84	147(1):57–84	NOUN
ap-7678	649	5	,	,	PUNCT
ap-7678	649	6	1983	1983	NUM
ap-7678	649	7	.	.	PUNCT
ap-7678	650	1	https://doi.org/10.1016/0003-4916(83)90067-2	https://doi.org/10.1016/0003-4916(83)90067-2	NOUN
ap-7678	650	2	.	.	PUNCT
ap-7678	651	1	[	[	X
ap-7678	651	2	13	13	NUM
ap-7678	651	3	]	]	PUNCT
ap-7678	651	4	j.	j.	PROPN
ap-7678	651	5	a.	a.	PROPN
ap-7678	651	6	shohat	shohat	PROPN
ap-7678	651	7	,	,	PUNCT
ap-7678	651	8	j.	j.	PROPN
ap-7678	651	9	d.	d.	PROPN
ap-7678	651	10	tamarkin	tamarkin	PROPN
ap-7678	651	11	.	.	PUNCT
ap-7678	652	1	the	the	DET
ap-7678	652	2	problem	problem	NOUN
ap-7678	652	3	of	of	ADP
ap-7678	652	4	moments	moment	NOUN
ap-7678	652	5	.	.	PUNCT
ap-7678	653	1	american	american	PROPN
ap-7678	653	2	mathematical	mathematical	PROPN
ap-7678	653	3	society	society	NOUN
ap-7678	653	4	,	,	PUNCT
ap-7678	653	5	providence	providence	NOUN
ap-7678	653	6	,	,	PUNCT
ap-7678	653	7	ri	ri	PROPN
ap-7678	653	8	,	,	PUNCT
ap-7678	653	9	1963	1963	NUM
ap-7678	653	10	.	.	PUNCT
ap-7678	654	1	[	[	X
ap-7678	654	2	14	14	NUM
ap-7678	654	3	]	]	X
ap-7678	654	4	c.	c.	PROPN
ap-7678	654	5	r.	r.	PROPN
ap-7678	654	6	handy	handy	PROPN
ap-7678	654	7	,	,	PUNCT
ap-7678	654	8	d.	d.	PROPN
ap-7678	654	9	bessis	bessis	NOUN
ap-7678	654	10	.	.	PUNCT
ap-7678	655	1	rapidly	rapidly	ADV
ap-7678	655	2	convergent	convergent	VERB
ap-7678	655	3	lower	low	ADJ
ap-7678	655	4	bounds	bound	NOUN
ap-7678	655	5	for	for	ADP
ap-7678	655	6	the	the	DET
ap-7678	655	7	schrödinger	schrödinger	ADJ
ap-7678	655	8	-	-	PUNCT
ap-7678	655	9	equation	equation	NOUN
ap-7678	655	10	ground	ground	NOUN
ap-7678	655	11	-	-	PUNCT
ap-7678	655	12	state	state	NOUN
ap-7678	655	13	energy	energy	NOUN
ap-7678	655	14	.	.	PUNCT
ap-7678	656	1	physical	physical	ADJ
ap-7678	656	2	review	review	NOUN
ap-7678	656	3	letters	letter	VERB
ap-7678	656	4	55:931–934	55:931–934	PROPN
ap-7678	656	5	,	,	PUNCT
ap-7678	656	6	1985	1985	NUM
ap-7678	656	7	.	.	PUNCT
ap-7678	657	1	https://doi.org/10.1103/physrevlett.55.931	https://doi.org/10.1103/physrevlett.55.931	NOUN
ap-7678	657	2	.	.	PUNCT
ap-7678	658	1	[	[	X
ap-7678	658	2	15	15	NUM
ap-7678	658	3	]	]	X
ap-7678	658	4	c.	c.	PROPN
ap-7678	658	5	r.	r.	PROPN
ap-7678	658	6	handy	handy	PROPN
ap-7678	658	7	,	,	PUNCT
ap-7678	658	8	d.	d.	PROPN
ap-7678	658	9	bessis	bessis	PROPN
ap-7678	658	10	,	,	PUNCT
ap-7678	658	11	g.	g.	PROPN
ap-7678	658	12	sigismondi	sigismondi	PROPN
ap-7678	658	13	,	,	PUNCT
ap-7678	658	14	t.	t.	PROPN
ap-7678	658	15	d.	d.	PROPN
ap-7678	658	16	morley	morley	PROPN
ap-7678	658	17	.	.	PUNCT
ap-7678	659	1	rapidly	rapidly	ADV
ap-7678	659	2	converging	converge	VERB
ap-7678	659	3	bounds	bound	NOUN
ap-7678	659	4	for	for	ADP
ap-7678	659	5	the	the	DET
ap-7678	659	6	ground	ground	NOUN
ap-7678	659	7	-	-	PUNCT
ap-7678	659	8	state	state	NOUN
ap-7678	659	9	energy	energy	NOUN
ap-7678	659	10	of	of	ADP
ap-7678	659	11	hydrogenic	hydrogenic	ADJ
ap-7678	659	12	atoms	atom	NOUN
ap-7678	659	13	in	in	ADP
ap-7678	659	14	superstrong	superstrong	ADJ
ap-7678	659	15	magnetic	magnetic	ADJ
ap-7678	659	16	fields	field	NOUN
ap-7678	659	17	.	.	PUNCT
ap-7678	660	1	physical	physical	ADJ
ap-7678	660	2	review	review	NOUN
ap-7678	660	3	letters	letter	VERB
ap-7678	660	4	60:253–256	60:253–256	PROPN
ap-7678	660	5	,	,	PUNCT
ap-7678	660	6	1988	1988	NUM
ap-7678	660	7	.	.	PUNCT
ap-7678	661	1	https://doi.org/10.1103/physrevlett.60.253	https://doi.org/10.1103/physrevlett.60.253	INTJ
ap-7678	661	2	.	.	PUNCT
ap-7678	662	1	[	[	X
ap-7678	662	2	16	16	NUM
ap-7678	662	3	]	]	X
ap-7678	662	4	y.	y.	PROPN
ap-7678	662	5	p.	p.	PROPN
ap-7678	662	6	kravchenko	kravchenko	PROPN
ap-7678	662	7	,	,	PUNCT
ap-7678	662	8	m.	m.	NOUN
ap-7678	662	9	a.	a.	PROPN
ap-7678	662	10	liberman	liberman	PROPN
ap-7678	662	11	,	,	PUNCT
ap-7678	662	12	b.	b.	PROPN
ap-7678	662	13	johansson	johansson	PROPN
ap-7678	662	14	.	.	PUNCT
ap-7678	663	1	exact	exact	ADJ
ap-7678	663	2	solution	solution	NOUN
ap-7678	663	3	for	for	ADP
ap-7678	663	4	a	a	DET
ap-7678	663	5	hydrogen	hydrogen	NOUN
ap-7678	663	6	atom	atom	NOUN
ap-7678	663	7	in	in	ADP
ap-7678	663	8	a	a	DET
ap-7678	663	9	magnetic	magnetic	ADJ
ap-7678	663	10	field	field	NOUN
ap-7678	663	11	of	of	ADP
ap-7678	663	12	arbitrary	arbitrary	ADJ
ap-7678	663	13	strength	strength	NOUN
ap-7678	663	14	.	.	PUNCT
ap-7678	664	1	physical	physical	ADJ
ap-7678	664	2	review	review	NOUN
ap-7678	664	3	a	a	DET
ap-7678	664	4	54:287–305	54:287–305	PROPN
ap-7678	664	5	,	,	PUNCT
ap-7678	664	6	1996	1996	NUM
ap-7678	664	7	.	.	PUNCT
ap-7678	665	1	https://doi.org/10.1103/physreva.54.287	https://doi.org/10.1103/physreva.54.287	PROPN
ap-7678	665	2	.	.	PUNCT
ap-7678	666	1	[	[	X
ap-7678	666	2	17	17	NUM
ap-7678	666	3	]	]	X
ap-7678	666	4	c.	c.	PROPN
ap-7678	666	5	schimeczek	schimeczek	PROPN
ap-7678	666	6	,	,	PUNCT
ap-7678	666	7	g.	g.	PROPN
ap-7678	666	8	wunner	wunner	PROPN
ap-7678	666	9	.	.	PUNCT
ap-7678	667	1	accurate	accurate	ADJ
ap-7678	667	2	2d	2d	PROPN
ap-7678	667	3	finite	finite	PROPN
ap-7678	667	4	element	element	NOUN
ap-7678	667	5	calculations	calculation	NOUN
ap-7678	667	6	for	for	ADP
ap-7678	667	7	hydrogen	hydrogen	NOUN
ap-7678	667	8	in	in	ADP
ap-7678	667	9	magnetic	magnetic	ADJ
ap-7678	667	10	fields	field	NOUN
ap-7678	667	11	of	of	ADP
ap-7678	667	12	arbitrary	arbitrary	ADJ
ap-7678	667	13	strength	strength	NOUN
ap-7678	667	14	.	.	PUNCT
ap-7678	668	1	computer	computer	NOUN
ap-7678	668	2	physics	physics	NOUN
ap-7678	668	3	communications	communication	NOUN
ap-7678	668	4	185(2):614–621	185(2):614–621	NUM
ap-7678	668	5	,	,	PUNCT
ap-7678	668	6	2014	2014	NUM
ap-7678	668	7	.	.	PUNCT
ap-7678	669	1	https://doi.org/10.1016/j.cpc.2013.09.023	https://doi.org/10.1016/j.cpc.2013.09.023	PROPN
ap-7678	669	2	.	.	PUNCT
ap-7678	670	1	[	[	X
ap-7678	670	2	18	18	NUM
ap-7678	670	3	]	]	PUNCT
ap-7678	670	4	a.	a.	NOUN
ap-7678	670	5	grossmann	grossmann	PROPN
ap-7678	670	6	,	,	PUNCT
ap-7678	670	7	j.	j.	PROPN
ap-7678	670	8	morlet	morlet	PROPN
ap-7678	670	9	.	.	PUNCT
ap-7678	671	1	decomposition	decomposition	NOUN
ap-7678	671	2	of	of	ADP
ap-7678	671	3	hardy	hardy	ADJ
ap-7678	671	4	functions	function	NOUN
ap-7678	671	5	into	into	ADP
ap-7678	671	6	square	square	ADJ
ap-7678	671	7	integrable	integrable	ADJ
ap-7678	671	8	wavelets	wavelet	NOUN
ap-7678	671	9	of	of	ADP
ap-7678	671	10	constant	constant	ADJ
ap-7678	671	11	shape	shape	NOUN
ap-7678	671	12	.	.	PUNCT
ap-7678	672	1	siam	siam	PROPN
ap-7678	672	2	journal	journal	PROPN
ap-7678	672	3	on	on	ADP
ap-7678	672	4	mathematical	mathematical	ADJ
ap-7678	672	5	analysis	analysis	NOUN
ap-7678	672	6	15(4):723–736	15(4):723–736	NUM
ap-7678	672	7	,	,	PUNCT
ap-7678	672	8	1984	1984	NUM
ap-7678	672	9	.	.	PUNCT
ap-7678	673	1	https://doi.org/10.1137/0515056	https://doi.org/10.1137/0515056	NOUN
ap-7678	673	2	.	.	PUNCT
ap-7678	674	1	[	[	X
ap-7678	674	2	19	19	NUM
ap-7678	674	3	]	]	X
ap-7678	674	4	c.	c.	PROPN
ap-7678	674	5	handy	handy	PROPN
ap-7678	674	6	,	,	PUNCT
ap-7678	674	7	r.	r.	PROPN
ap-7678	674	8	murenzi	murenzi	PROPN
ap-7678	674	9	.	.	PUNCT
ap-7678	675	1	moment	moment	NOUN
ap-7678	675	2	-	-	PUNCT
ap-7678	675	3	wavelet	wavelet	NOUN
ap-7678	675	4	quantization	quantization	NOUN
ap-7678	675	5	:	:	PUNCT
ap-7678	675	6	a	a	DET
ap-7678	675	7	first	first	ADJ
ap-7678	675	8	principles	principle	NOUN
ap-7678	675	9	analysis	analysis	NOUN
ap-7678	675	10	of	of	ADP
ap-7678	675	11	quantum	quantum	ADJ
ap-7678	675	12	mechanics	mechanic	NOUN
ap-7678	675	13	through	through	ADP
ap-7678	675	14	continuous	continuous	ADJ
ap-7678	675	15	wavelet	wavelet	NOUN
ap-7678	675	16	transform	transform	NOUN
ap-7678	675	17	theory	theory	NOUN
ap-7678	675	18	.	.	PUNCT
ap-7678	676	1	physics	physics	NOUN
ap-7678	676	2	letters	letter	NOUN
ap-7678	676	3	a	a	DET
ap-7678	676	4	248(1):7–15	248(1):7–15	NUM
ap-7678	676	5	,	,	PUNCT
ap-7678	676	6	1998	1998	NUM
ap-7678	676	7	.	.	PUNCT
ap-7678	677	1	https://doi.org/10.1016/s0375-9601(98)00645-8	https://doi.org/10.1016/s0375-9601(98)00645-8	NOUN
ap-7678	677	2	.	.	PUNCT
ap-7678	678	1	[	[	X
ap-7678	678	2	20	20	NUM
ap-7678	678	3	]	]	X
ap-7678	678	4	c.	c.	PROPN
ap-7678	678	5	r.	r.	PROPN
ap-7678	678	6	handy	handy	PROPN
ap-7678	678	7	.	.	PUNCT
ap-7678	679	1	singular	singular	NOUN
ap-7678	679	2	-	-	PUNCT
ap-7678	679	3	perturbation	perturbation	NOUN
ap-7678	679	4	–	–	PUNCT
ap-7678	679	5	strong	strong	ADJ
ap-7678	679	6	-	-	PUNCT
ap-7678	679	7	coupling	couple	VERB
ap-7678	679	8	field	field	NOUN
ap-7678	679	9	theory	theory	NOUN
ap-7678	679	10	and	and	CCONJ
ap-7678	679	11	the	the	DET
ap-7678	679	12	moments	moment	NOUN
ap-7678	679	13	problem	problem	NOUN
ap-7678	679	14	.	.	PUNCT
ap-7678	680	1	physical	physical	ADJ
ap-7678	680	2	review	review	PROPN
ap-7678	680	3	d	d	PROPN
ap-7678	680	4	24:378–383	24:378–383	NUM
ap-7678	680	5	,	,	PUNCT
ap-7678	680	6	1981	1981	NUM
ap-7678	680	7	.	.	PUNCT
ap-7678	681	1	https://doi.org/10.1103/physrevd.24.378	https://doi.org/10.1103/physrevd.24.378	NOUN
ap-7678	681	2	.	.	PUNCT
ap-7678	682	1	[	[	X
ap-7678	682	2	21	21	NUM
ap-7678	682	3	]	]	X
ap-7678	682	4	s.	s.	PROPN
ap-7678	682	5	boyd	boyd	PROPN
ap-7678	682	6	,	,	PUNCT
ap-7678	682	7	l.	l.	PROPN
ap-7678	682	8	vandenberghe	vandenberghe	PROPN
ap-7678	682	9	.	.	PUNCT
ap-7678	683	1	convex	convex	PROPN
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ap-7678	683	3	.	.	PUNCT
ap-7678	684	1	cambridge	cambridge	PROPN
ap-7678	684	2	university	university	PROPN
ap-7678	684	3	press	press	PROPN
ap-7678	684	4	,	,	PUNCT
ap-7678	684	5	new	new	PROPN
ap-7678	684	6	york	york	PROPN
ap-7678	684	7	,	,	PUNCT
ap-7678	684	8	2004	2004	NUM
ap-7678	684	9	.	.	PUNCT
ap-7678	685	1	[	[	X
ap-7678	685	2	22	22	NUM
ap-7678	685	3	]	]	PUNCT
ap-7678	685	4	j.	j.	PROPN
ap-7678	685	5	b.	b.	PROPN
ap-7678	685	6	lasserre	lasserre	PROPN
ap-7678	685	7	.	.	PUNCT
ap-7678	686	1	moments	moment	NOUN
ap-7678	686	2	,	,	PUNCT
ap-7678	686	3	positive	positive	ADJ
ap-7678	686	4	polynomials	polynomial	NOUN
ap-7678	686	5	and	and	CCONJ
ap-7678	686	6	their	their	PRON
ap-7678	686	7	applications	application	NOUN
ap-7678	686	8	.	.	PUNCT
ap-7678	687	1	imperial	imperial	ADJ
ap-7678	687	2	college	college	PROPN
ap-7678	687	3	press	press	PROPN
ap-7678	687	4	,	,	PUNCT
ap-7678	687	5	london	london	PROPN
ap-7678	687	6	,	,	PUNCT
ap-7678	687	7	2010	2010	NUM
ap-7678	687	8	.	.	PUNCT
ap-7678	688	1	[	[	X
ap-7678	688	2	23	23	X
ap-7678	688	3	]	]	X
ap-7678	688	4	v.	v.	PROPN
ap-7678	688	5	chvátal	chvátal	PROPN
ap-7678	688	6	.	.	PUNCT
ap-7678	689	1	linear	linear	PROPN
ap-7678	689	2	programming	programming	PROPN
ap-7678	689	3	.	.	PUNCT
ap-7678	690	1	freeman	freeman	PROPN
ap-7678	690	2	,	,	PUNCT
ap-7678	690	3	new	new	PROPN
ap-7678	690	4	york	york	PROPN
ap-7678	690	5	,	,	PUNCT
ap-7678	690	6	1983	1983	NUM
ap-7678	690	7	.	.	PUNCT
ap-7678	691	1	[	[	X
ap-7678	691	2	24	24	NUM
ap-7678	691	3	]	]	PUNCT
ap-7678	691	4	m.	m.	NOUN
ap-7678	691	5	reed	reed	PROPN
ap-7678	691	6	,	,	PUNCT
ap-7678	691	7	b.	b.	PROPN
ap-7678	691	8	simon	simon	PROPN
ap-7678	691	9	.	.	PUNCT
ap-7678	692	1	methods	method	NOUN
ap-7678	692	2	of	of	ADP
ap-7678	692	3	modern	modern	ADJ
ap-7678	692	4	mathematical	mathematical	ADJ
ap-7678	692	5	physics	physics	NOUN
ap-7678	692	6	.	.	PUNCT
ap-7678	693	1	vol	vol	NOUN
ap-7678	693	2	.	.	PROPN
ap-7678	694	1	1	1	NUM
ap-7678	694	2	-	-	SYM
ap-7678	694	3	3	3	NUM
ap-7678	694	4	.	.	PUNCT
ap-7678	694	5	academic	academic	ADJ
ap-7678	694	6	press	press	NOUN
ap-7678	694	7	,	,	PUNCT
ap-7678	694	8	new	new	PROPN
ap-7678	694	9	york	york	PROPN
ap-7678	694	10	,	,	PUNCT
ap-7678	694	11	1972	1972	NUM
ap-7678	694	12	.	.	PUNCT
ap-7678	695	1	https	https	NOUN
ap-7678	695	2	:	:	PUNCT
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ap-7678	696	2	/	/	SYM
ap-7678	696	3	b978	b978	NUM
ap-7678	696	4	-	-	PUNCT
ap-7678	696	5	0	0	NUM
ap-7678	696	6	-	-	PUNCT
ap-7678	696	7	12	12	NUM
ap-7678	696	8	-	-	PUNCT
ap-7678	696	9	585001	585001	NUM
ap-7678	696	10	-	-	PUNCT
ap-7678	696	11	8.x5001	8.x5001	NUM
ap-7678	696	12	-	-	PUNCT
ap-7678	696	13	6	6	NUM
ap-7678	696	14	.	.	PUNCT
ap-7678	697	1	[	[	X
ap-7678	697	2	25	25	NUM
ap-7678	697	3	]	]	X
ap-7678	697	4	c.	c.	PROPN
ap-7678	697	5	r.	r.	PROPN
ap-7678	697	6	handy	handy	PROPN
ap-7678	697	7	,	,	PUNCT
ap-7678	697	8	d.	d.	PROPN
ap-7678	697	9	khan	khan	PROPN
ap-7678	697	10	,	,	PUNCT
ap-7678	697	11	x.-q	x.-q	PROPN
ap-7678	697	12	.	.	PUNCT
ap-7678	698	1	wang	wang	PROPN
ap-7678	698	2	,	,	PUNCT
ap-7678	698	3	c.	c.	PROPN
ap-7678	698	4	j.	j.	PROPN
ap-7678	698	5	tymczak	tymczak	PROPN
ap-7678	698	6	.	.	PUNCT
ap-7678	699	1	multiscale	multiscale	PROPN
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ap-7678	699	3	function	function	VERB
ap-7678	699	4	analysis	analysis	NOUN
ap-7678	699	5	of	of	ADP
ap-7678	699	6	the	the	DET
ap-7678	699	7	pt	pt	PROPN
ap-7678	699	8	symmetry	symmetry	NOUN
ap-7678	699	9	breaking	break	VERB
ap-7678	699	10	solutions	solution	NOUN
ap-7678	699	11	for	for	ADP
ap-7678	699	12	p	p	NOUN
ap-7678	699	13	2+ix3+iαx	2+ix3+iαx	NUM
ap-7678	699	14	hamiltonian	hamiltonian	NOUN
ap-7678	699	15	.	.	PUNCT
ap-7678	700	1	journal	journal	PROPN
ap-7678	700	2	of	of	ADP
ap-7678	700	3	physics	physics	PROPN
ap-7678	700	4	a	a	PRON
ap-7678	700	5	:	:	PUNCT
ap-7678	700	6	mathematical	mathematical	ADJ
ap-7678	700	7	and	and	CCONJ
ap-7678	700	8	general	general	ADJ
ap-7678	700	9	34(27):5593–5602	34(27):5593–5602	NUM
ap-7678	700	10	,	,	PUNCT
ap-7678	700	11	2001	2001	NUM
ap-7678	700	12	.	.	PUNCT
ap-7678	701	1	https://doi.org/10.1088/0305-4470/34/27/309	https://doi.org/10.1088/0305-4470/34/27/309	PROPN
ap-7678	701	2	.	.	PUNCT
ap-7678	702	1	[	[	X
ap-7678	702	2	26	26	NUM
ap-7678	702	3	]	]	X
ap-7678	702	4	e.	e.	PROPN
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ap-7678	702	6	,	,	PUNCT
ap-7678	702	7	d.	d.	PROPN
ap-7678	702	8	t.	t.	PROPN
ap-7678	702	9	trinh	trinh	PROPN
ap-7678	702	10	.	.	PUNCT
ap-7678	703	1	spectral	spectral	ADJ
ap-7678	703	2	analysis	analysis	NOUN
ap-7678	703	3	of	of	ADP
ap-7678	703	4	the	the	DET
ap-7678	703	5	complex	complex	ADJ
ap-7678	703	6	cubic	cubic	ADJ
ap-7678	703	7	oscillator	oscillator	NOUN
ap-7678	703	8	.	.	PUNCT
ap-7678	704	1	journal	journal	PROPN
ap-7678	704	2	of	of	ADP
ap-7678	704	3	physics	physics	PROPN
ap-7678	704	4	a	a	PRON
ap-7678	704	5	:	:	PUNCT
ap-7678	704	6	mathematical	mathematical	ADJ
ap-7678	704	7	and	and	CCONJ
ap-7678	704	8	general	general	ADJ
ap-7678	704	9	33(48):8771–8796	33(48):8771–8796	NUM
ap-7678	704	10	,	,	PUNCT
ap-7678	704	11	2000	2000	NUM
ap-7678	704	12	.	.	PUNCT
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ap-7678	705	2	.	.	PUNCT
ap-7678	706	1	[	[	X
ap-7678	706	2	27	27	NUM
ap-7678	706	3	]	]	X
ap-7678	706	4	c.	c.	PROPN
ap-7678	706	5	r.	r.	PROPN
ap-7678	706	6	handy	handy	PROPN
ap-7678	706	7	.	.	PUNCT
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ap-7678	707	2	converging	converge	VERB
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ap-7678	707	4	bounds	bound	NOUN
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ap-7678	707	7	discrete	discrete	ADJ
ap-7678	707	8	states	state	NOUN
ap-7678	707	9	of	of	ADP
ap-7678	707	10	the	the	DET
ap-7678	707	11	ix3	ix3	PROPN
ap-7678	707	12	non	non	ADJ
ap-7678	707	13	-	-	ADJ
ap-7678	707	14	hermitian	hermitian	ADJ
ap-7678	707	15	potential	potential	NOUN
ap-7678	707	16	.	.	PUNCT
ap-7678	708	1	journal	journal	PROPN
ap-7678	708	2	of	of	ADP
ap-7678	708	3	physics	physics	PROPN
ap-7678	708	4	a	a	PRON
ap-7678	708	5	:	:	PUNCT
ap-7678	708	6	mathematical	mathematical	ADJ
ap-7678	708	7	and	and	CCONJ
ap-7678	708	8	general	general	ADJ
ap-7678	708	9	34(19):l271	34(19):l271	NUM
ap-7678	708	10	–	–	PUNCT
ap-7678	708	11	l277	l277	PROPN
ap-7678	708	12	,	,	PUNCT
ap-7678	708	13	2001	2001	NUM
ap-7678	708	14	.	.	PUNCT
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ap-7678	711	1	https://doi.org/10.1088/1751-8113/46/13/135202	https://doi.org/10.1088/1751-8113/46/13/135202	NOUN
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ap-7678	711	12	https://doi.org/10.1016/b978-0-12-585001-8.x5001-6	https://doi.org/10.1016/b978-0-12-585001-8.x5001-6	X
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ap-7678	711	14	https://doi.org/10.1088/0305-4470/33/48/314	https://doi.org/10.1088/0305-4470/33/48/314	PROPN
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ap-7678	711	16	acta	acta	PROPN
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ap-7678	711	21	1	1	NUM
ap-7678	711	22	introduction	introduction	NOUN
ap-7678	711	23	1.1	1.1	NUM
ap-7678	711	24	singular	singular	NOUN
ap-7678	711	25	-	-	PUNCT
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ap-7678	711	27	strong	strong	ADJ
ap-7678	711	28	coupling	coupling	NOUN
ap-7678	711	29	problems	problem	NOUN
ap-7678	711	30	and	and	CCONJ
ap-7678	711	31	moment	moment	NOUN
ap-7678	711	32	representations	representation	VERB
ap-7678	711	33	2	2	NUM
ap-7678	711	34	the	the	DET
ap-7678	711	35	moment	moment	NOUN
ap-7678	711	36	equation	equation	NOUN
ap-7678	711	37	representation	representation	NOUN
ap-7678	711	38	3	3	NUM
ap-7678	711	39	generating	generate	VERB
ap-7678	711	40	eigenenergy	eigenenergy	NOUN
ap-7678	711	41	bounds	bound	NOUN
ap-7678	711	42	within	within	ADP
ap-7678	711	43	a	a	DET
ap-7678	711	44	moments	moment	NOUN
ap-7678	711	45	'	'	PART
ap-7678	711	46	representation	representation	NOUN
ap-7678	711	47	4	4	NUM
ap-7678	711	48	the	the	DET
ap-7678	711	49	eigenvalue	eigenvalue	PROPN
ap-7678	711	50	moment	moment	NOUN
ap-7678	711	51	method	method	NOUN
ap-7678	711	52	5	5	NUM
ap-7678	711	53	the	the	DET
ap-7678	711	54	orthonormal	orthonormal	ADJ
ap-7678	711	55	polynomial	polynomial	ADJ
ap-7678	711	56	projection	projection	NOUN
ap-7678	711	57	quantization	quantization	NOUN
ap-7678	711	58	formalism	formalism	NOUN
ap-7678	711	59	5.1	5.1	NUM
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ap-7678	711	61	oppq	oppq	PROPN
ap-7678	711	62	non	non	ADJ
ap-7678	711	63	-	-	ADJ
ap-7678	711	64	orthogonal	orthogonal	ADJ
ap-7678	711	65	basis	basis	NOUN
ap-7678	711	66	expansion	expansion	NOUN
ap-7678	711	67	5.2	5.2	NUM
ap-7678	711	68	the	the	DET
ap-7678	711	69	oppq	oppq	ADJ
ap-7678	711	70	quantization	quantization	NOUN
ap-7678	711	71	condition	condition	NOUN
ap-7678	711	72	5.2.1	5.2.1	NUM
ap-7678	711	73	selection	selection	NOUN
ap-7678	711	74	of	of	ADP
ap-7678	711	75	the	the	DET
ap-7678	711	76	oppq	oppq	ADJ
ap-7678	711	77	weight	weight	NOUN
ap-7678	711	78	5.2.2	5.2.2	NUM
ap-7678	711	79	use	use	NOUN
ap-7678	711	80	of	of	ADP
ap-7678	711	81	the	the	DET
ap-7678	711	82	ground	ground	NOUN
ap-7678	711	83	state	state	NOUN
ap-7678	711	84	as	as	ADP
ap-7678	711	85	a	a	DET
ap-7678	711	86	weight	weight	NOUN
ap-7678	711	87	6	6	NUM
ap-7678	711	88	the	the	DET
ap-7678	711	89	oppq	oppq	NOUN
ap-7678	711	90	-	-	PUNCT
ap-7678	711	91	approximation	approximation	NOUN
ap-7678	711	92	method	method	NOUN
ap-7678	711	93	(	(	PUNCT
ap-7678	711	94	oppq	oppq	PROPN
ap-7678	711	95	-	-	PUNCT
ap-7678	711	96	am	am	NOUN
ap-7678	711	97	)	)	PUNCT
ap-7678	711	98	7	7	NUM
ap-7678	711	99	the	the	DET
ap-7678	711	100	oppq	oppq	NOUN
ap-7678	711	101	-	-	PUNCT
ap-7678	711	102	bounding	bound	VERB
ap-7678	711	103	method	method	NOUN
ap-7678	711	104	8	8	NUM
ap-7678	711	105	the	the	DET
ap-7678	711	106	oppq	oppq	NOUN
ap-7678	711	107	-	-	PUNCT
ap-7678	711	108	bounding	bound	VERB
ap-7678	711	109	method	method	NOUN
ap-7678	711	110	9	9	NUM
ap-7678	711	111	the	the	DET
ap-7678	711	112	sextic	sextic	ADJ
ap-7678	711	113	anharmonic	anharmonic	ADJ
ap-7678	711	114	double	double	ADJ
ap-7678	711	115	well	well	NOUN
ap-7678	711	116	potential	potential	ADJ
ap-7678	711	117	9.1	9.1	NUM
ap-7678	711	118	emm	emm	PROPN
ap-7678	711	119	9.2	9.2	NUM
ap-7678	711	120	emm-2	emm-2	NUM
ap-7678	711	121	9.3	9.3	NUM
ap-7678	711	122	emm	emm	NOUN
ap-7678	711	123	e	e	NOUN
ap-7678	711	124	-	-	NOUN
ap-7678	711	125	x44(x	x44(x	NUM
ap-7678	711	126	)	)	PUNCT
ap-7678	711	127	9.4	9.4	NUM
ap-7678	711	128	oppq	oppq	ADJ
ap-7678	711	129	analysis	analysis	NOUN
ap-7678	711	130	of	of	ADP
ap-7678	711	131	the	the	DET
ap-7678	711	132	(	(	PUNCT
ap-7678	711	133	ms	ms	NOUN
ap-7678	711	134	=	=	SYM
ap-7678	711	135	0	0	NUM
ap-7678	711	136	)	)	PUNCT
ap-7678	711	137	sextic	sextic	ADJ
ap-7678	711	138	anharmonic	anharmonic	ADJ
ap-7678	711	139	double	double	ADJ
ap-7678	711	140	well	well	NOUN
ap-7678	711	141	oscillator	oscillator	NOUN
ap-7678	711	142	9.5	9.5	NUM
ap-7678	711	143	oppq	oppq	ADJ
ap-7678	711	144	results	result	NOUN
ap-7678	711	145	for	for	ADP
ap-7678	711	146	the	the	DET
ap-7678	711	147	ms=0	ms=0	PROPN
ap-7678	711	148	,	,	PUNCT
ap-7678	711	149	sextic	sextic	ADJ
ap-7678	711	150	anharmonic	anharmonic	ADJ
ap-7678	711	151	oscillator	oscillator	NOUN
ap-7678	711	152	9.5.1	9.5.1	NUM
ap-7678	711	153	generating	generating	NOUN
ap-7678	711	154	bounds	bound	NOUN
ap-7678	711	155	for	for	ADP
ap-7678	711	156	e8	e8	PROPN
ap-7678	711	157	9.6	9.6	NUM
ap-7678	711	158	comparison	comparison	NOUN
ap-7678	711	159	with	with	ADP
ap-7678	711	160	emm	emm	PROPN
ap-7678	711	161	bounds	bound	VERB
ap-7678	711	162	9.7	9.7	NUM
ap-7678	711	163	using	use	VERB
ap-7678	711	164	the	the	DET
ap-7678	711	165	ground	ground	NOUN
ap-7678	711	166	state	state	NOUN
ap-7678	711	167	as	as	ADP
ap-7678	711	168	a	a	DET
ap-7678	711	169	weight	weight	NOUN
ap-7678	711	170	9.8	9.8	NUM
ap-7678	711	171	emm	emm	PROPN
ap-7678	711	172	results	result	NOUN
ap-7678	711	173	for	for	ADP
ap-7678	711	174	the	the	DET
ap-7678	711	175	probability	probability	NOUN
ap-7678	711	176	density	density	NOUN
ap-7678	711	177	10	10	NUM
ap-7678	711	178	a	a	DET
ap-7678	711	179	pt	pt	ADJ
ap-7678	711	180	-	-	PUNCT
ap-7678	711	181	symmetry	symmetry	NOUN
ap-7678	711	182	breaking	breaking	NOUN
ap-7678	711	183	,	,	PUNCT
ap-7678	711	184	non	non	ADJ
ap-7678	711	185	-	-	ADJ
ap-7678	711	186	hermitian	hermitian	ADJ
ap-7678	711	187	,	,	PUNCT
ap-7678	711	188	problem	problem	NOUN
ap-7678	711	189	11	11	NUM
ap-7678	711	190	conclusions	conclusion	NOUN
ap-7678	711	191	acknowledgements	acknowledgement	NOUN
ap-7678	711	192	references	reference	NOUN