id	sid	tid	token	lemma	pos
ap-4589	1	1	acta	acta	PROPN
ap-4589	1	2	polytechnica	polytechnica	PROPN
ap-4589	1	3	doi:10.14311	doi:10.14311	PROPN
ap-4589	1	4	/	/	PUNCT
ap-4589	1	5	ap.2017.57.0446	ap.2017.57.0446	PROPN
ap-4589	1	6	acta	acta	PROPN
ap-4589	1	7	polytechnica	polytechnica	PROPN
ap-4589	1	8	57(6):446–453	57(6):446–453	PROPN
ap-4589	1	9	,	,	PUNCT
ap-4589	1	10	2017	2017	NUM
ap-4589	1	11	©	©	PROPN
ap-4589	1	12	czech	czech	PROPN
ap-4589	1	13	technical	technical	PROPN
ap-4589	1	14	university	university	PROPN
ap-4589	1	15	in	in	ADP
ap-4589	1	16	prague	prague	PROPN
ap-4589	1	17	,	,	PUNCT
ap-4589	1	18	2017	2017	NUM
ap-4589	1	19	available	available	ADJ
ap-4589	1	20	online	online	ADV
ap-4589	1	21	at	at	ADP
ap-4589	1	22	http://ojs.cvut.cz/ojs/index.php/ap	http://ojs.cvut.cz/ojs/index.php/ap	PROPN
ap-4589	1	23	algebraic	algebraic	ADJ
ap-4589	1	24	description	description	NOUN
ap-4589	1	25	of	of	ADP
ap-4589	1	26	shape	shape	NOUN
ap-4589	1	27	invariance	invariance	NOUN
ap-4589	1	28	revisited	revisit	VERB
ap-4589	1	29	satoshi	satoshi	PROPN
ap-4589	1	30	ohya	ohya	PROPN
ap-4589	1	31	institute	institute	PROPN
ap-4589	1	32	of	of	ADP
ap-4589	1	33	quantum	quantum	PROPN
ap-4589	1	34	science	science	NOUN
ap-4589	1	35	,	,	PUNCT
ap-4589	1	36	nihon	nihon	PROPN
ap-4589	1	37	university	university	PROPN
ap-4589	1	38	,	,	PUNCT
ap-4589	1	39	kanda	kanda	PROPN
ap-4589	1	40	-	-	PUNCT
ap-4589	1	41	surugadai	surugadai	PROPN
ap-4589	1	42	1	1	NUM
ap-4589	1	43	-	-	SYM
ap-4589	1	44	8	8	NUM
ap-4589	1	45	-	-	SYM
ap-4589	1	46	14	14	NUM
ap-4589	1	47	,	,	PUNCT
ap-4589	1	48	chiyoda	chiyoda	PROPN
ap-4589	1	49	,	,	PUNCT
ap-4589	1	50	tokyo	tokyo	PROPN
ap-4589	1	51	101	101	NUM
ap-4589	1	52	-	-	SYM
ap-4589	1	53	8308	8308	NUM
ap-4589	1	54	,	,	PUNCT
ap-4589	1	55	japan	japan	PROPN
ap-4589	1	56	correspondence	correspondence	NOUN
ap-4589	1	57	:	:	PUNCT
ap-4589	1	58	ohya@phys.cst.nihon-u.ac.jp	ohya@phys.cst.nihon-u.ac.jp	ADJ
ap-4589	1	59	abstract	abstract	NOUN
ap-4589	1	60	.	.	PUNCT
ap-4589	2	1	we	we	PRON
ap-4589	2	2	revisit	revisit	VERB
ap-4589	2	3	the	the	DET
ap-4589	2	4	algebraic	algebraic	ADJ
ap-4589	2	5	description	description	NOUN
ap-4589	2	6	of	of	ADP
ap-4589	2	7	shape	shape	NOUN
ap-4589	2	8	invariance	invariance	NOUN
ap-4589	2	9	method	method	NOUN
ap-4589	2	10	in	in	ADP
ap-4589	2	11	one	one	NUM
ap-4589	2	12	-	-	PUNCT
ap-4589	2	13	dimensional	dimensional	ADJ
ap-4589	2	14	quantum	quantum	ADJ
ap-4589	2	15	mechanics	mechanic	NOUN
ap-4589	2	16	.	.	PUNCT
ap-4589	3	1	in	in	ADP
ap-4589	3	2	this	this	DET
ap-4589	3	3	note	note	NOUN
ap-4589	3	4	we	we	PRON
ap-4589	3	5	focus	focus	VERB
ap-4589	3	6	on	on	ADP
ap-4589	3	7	four	four	NUM
ap-4589	3	8	particular	particular	ADJ
ap-4589	3	9	examples	example	NOUN
ap-4589	3	10	:	:	PUNCT
ap-4589	3	11	the	the	DET
ap-4589	3	12	kepler	kepler	PROPN
ap-4589	3	13	problem	problem	NOUN
ap-4589	3	14	in	in	ADP
ap-4589	3	15	flat	flat	ADJ
ap-4589	3	16	space	space	NOUN
ap-4589	3	17	,	,	PUNCT
ap-4589	3	18	the	the	DET
ap-4589	3	19	kepler	kepler	PROPN
ap-4589	3	20	problem	problem	NOUN
ap-4589	3	21	in	in	ADP
ap-4589	3	22	spherical	spherical	ADJ
ap-4589	3	23	space	space	NOUN
ap-4589	3	24	,	,	PUNCT
ap-4589	3	25	the	the	DET
ap-4589	3	26	kepler	kepler	PROPN
ap-4589	3	27	problem	problem	NOUN
ap-4589	3	28	in	in	ADP
ap-4589	3	29	hyperbolic	hyperbolic	ADJ
ap-4589	3	30	space	space	NOUN
ap-4589	3	31	,	,	PUNCT
ap-4589	3	32	and	and	CCONJ
ap-4589	3	33	the	the	DET
ap-4589	3	34	rosen	rosen	PROPN
ap-4589	3	35	–	–	PUNCT
ap-4589	3	36	morse	morse	ADJ
ap-4589	3	37	potential	potential	ADJ
ap-4589	3	38	problem	problem	NOUN
ap-4589	3	39	.	.	PUNCT
ap-4589	4	1	following	follow	VERB
ap-4589	4	2	the	the	DET
ap-4589	4	3	prescription	prescription	NOUN
ap-4589	4	4	given	give	VERB
ap-4589	4	5	by	by	ADP
ap-4589	4	6	gangopadhyaya	gangopadhyaya	PROPN
ap-4589	4	7	et	et	PROPN
ap-4589	4	8	al	al	PROPN
ap-4589	4	9	.	.	PROPN
ap-4589	4	10	,	,	PUNCT
ap-4589	4	11	we	we	PRON
ap-4589	4	12	first	first	ADV
ap-4589	4	13	introduce	introduce	VERB
ap-4589	4	14	certain	certain	ADJ
ap-4589	4	15	nonlinear	nonlinear	ADJ
ap-4589	4	16	algebraic	algebraic	ADJ
ap-4589	4	17	systems	system	NOUN
ap-4589	4	18	.	.	PUNCT
ap-4589	5	1	we	we	PRON
ap-4589	5	2	then	then	ADV
ap-4589	5	3	show	show	VERB
ap-4589	5	4	that	that	SCONJ
ap-4589	5	5	,	,	PUNCT
ap-4589	5	6	if	if	SCONJ
ap-4589	5	7	the	the	DET
ap-4589	5	8	model	model	NOUN
ap-4589	5	9	parameters	parameter	NOUN
ap-4589	5	10	are	be	AUX
ap-4589	5	11	appropriately	appropriately	ADV
ap-4589	5	12	quantized	quantize	VERB
ap-4589	5	13	,	,	PUNCT
ap-4589	5	14	the	the	DET
ap-4589	5	15	bound	bind	VERB
ap-4589	5	16	-	-	PUNCT
ap-4589	5	17	state	state	NOUN
ap-4589	5	18	problems	problem	NOUN
ap-4589	5	19	can	can	AUX
ap-4589	5	20	be	be	AUX
ap-4589	5	21	solved	solve	VERB
ap-4589	5	22	solely	solely	ADV
ap-4589	5	23	by	by	ADP
ap-4589	5	24	means	mean	NOUN
ap-4589	5	25	of	of	ADP
ap-4589	5	26	representation	representation	NOUN
ap-4589	5	27	theory	theory	NOUN
ap-4589	5	28	.	.	PUNCT
ap-4589	6	1	keywords	keyword	NOUN
ap-4589	6	2	:	:	PUNCT
ap-4589	6	3	exactly	exactly	ADV
ap-4589	6	4	solvable	solvable	ADJ
ap-4589	6	5	models	model	NOUN
ap-4589	6	6	;	;	PUNCT
ap-4589	6	7	shape	shape	NOUN
ap-4589	6	8	invariance	invariance	NOUN
ap-4589	6	9	;	;	PUNCT
ap-4589	6	10	representation	representation	NOUN
ap-4589	6	11	theory	theory	NOUN
ap-4589	6	12	.	.	PUNCT
ap-4589	7	1	1	1	X
ap-4589	7	2	.	.	X
ap-4589	7	3	introduction	introduction	NOUN
ap-4589	7	4	the	the	DET
ap-4589	7	5	purpose	purpose	NOUN
ap-4589	7	6	of	of	ADP
ap-4589	7	7	this	this	DET
ap-4589	7	8	note	note	NOUN
ap-4589	7	9	is	be	AUX
ap-4589	7	10	to	to	PART
ap-4589	7	11	revisit	revisit	VERB
ap-4589	7	12	a	a	DET
ap-4589	7	13	couple	couple	NOUN
ap-4589	7	14	of	of	ADP
ap-4589	7	15	one	one	NUM
ap-4589	7	16	-	-	PUNCT
ap-4589	7	17	dimensional	dimensional	ADJ
ap-4589	7	18	quantum	quantum	NOUN
ap-4589	7	19	-	-	ADJ
ap-4589	7	20	mechanical	mechanical	ADJ
ap-4589	7	21	bound	bind	VERB
ap-4589	7	22	-	-	PUNCT
ap-4589	7	23	state	state	NOUN
ap-4589	7	24	problems	problem	NOUN
ap-4589	7	25	that	that	PRON
ap-4589	7	26	can	can	AUX
ap-4589	7	27	be	be	AUX
ap-4589	7	28	solved	solve	VERB
ap-4589	7	29	exactly	exactly	ADV
ap-4589	7	30	.	.	PUNCT
ap-4589	8	1	in	in	ADP
ap-4589	8	2	this	this	DET
ap-4589	8	3	note	note	NOUN
ap-4589	8	4	we	we	PRON
ap-4589	8	5	shall	shall	AUX
ap-4589	8	6	focus	focus	VERB
ap-4589	8	7	on	on	ADP
ap-4589	8	8	four	four	NUM
ap-4589	8	9	particular	particular	ADJ
ap-4589	8	10	examples	example	NOUN
ap-4589	8	11	:	:	PUNCT
ap-4589	8	12	the	the	DET
ap-4589	8	13	kepler	kepler	PROPN
ap-4589	8	14	problem	problem	NOUN
ap-4589	8	15	in	in	ADP
ap-4589	8	16	flat	flat	ADJ
ap-4589	8	17	space	space	NOUN
ap-4589	8	18	,	,	PUNCT
ap-4589	8	19	the	the	DET
ap-4589	8	20	kepler	kepler	PROPN
ap-4589	8	21	problem	problem	NOUN
ap-4589	8	22	in	in	ADP
ap-4589	8	23	spherical	spherical	ADJ
ap-4589	8	24	space	space	NOUN
ap-4589	9	1	[	[	X
ap-4589	9	2	1–3	1–3	NOUN
ap-4589	9	3	]	]	X
ap-4589	9	4	,	,	PUNCT
ap-4589	9	5	the	the	DET
ap-4589	9	6	kepler	kepler	PROPN
ap-4589	9	7	problem	problem	NOUN
ap-4589	9	8	in	in	ADP
ap-4589	9	9	hyperbolic	hyperbolic	ADJ
ap-4589	9	10	space	space	NOUN
ap-4589	9	11	[	[	X
ap-4589	9	12	4	4	NUM
ap-4589	9	13	,	,	PUNCT
ap-4589	9	14	5	5	NUM
ap-4589	9	15	]	]	PUNCT
ap-4589	9	16	,	,	PUNCT
ap-4589	9	17	and	and	CCONJ
ap-4589	9	18	the	the	DET
ap-4589	9	19	rosen	rosen	PROPN
ap-4589	9	20	–	–	PUNCT
ap-4589	9	21	morse	morse	ADJ
ap-4589	9	22	potential	potential	ADJ
ap-4589	9	23	problem	problem	NOUN
ap-4589	10	1	[	[	X
ap-4589	10	2	6	6	NUM
ap-4589	10	3	,	,	PUNCT
ap-4589	10	4	7	7	NUM
ap-4589	10	5	]	]	PUNCT
ap-4589	10	6	,	,	PUNCT
ap-4589	10	7	all	all	PRON
ap-4589	10	8	of	of	ADP
ap-4589	10	9	whose	whose	DET
ap-4589	10	10	bound	bind	VERB
ap-4589	10	11	-	-	PUNCT
ap-4589	10	12	state	state	NOUN
ap-4589	10	13	spectra	spectra	NOUN
ap-4589	10	14	are	be	AUX
ap-4589	10	15	known	know	VERB
ap-4589	10	16	to	to	PART
ap-4589	10	17	be	be	AUX
ap-4589	10	18	exactly	exactly	ADV
ap-4589	10	19	calculable	calculable	ADJ
ap-4589	10	20	.	.	PUNCT
ap-4589	11	1	hamiltonians	hamiltonian	NOUN
ap-4589	11	2	of	of	ADP
ap-4589	11	3	these	these	DET
ap-4589	11	4	problems1	problems1	NOUN
ap-4589	11	5	are	be	AUX
ap-4589	11	6	respectively	respectively	ADV
ap-4589	11	7	given	give	VERB
ap-4589	11	8	by	by	ADP
ap-4589	11	9	hkepler	hkepler	NOUN
ap-4589	11	10	=	=	SYM
ap-4589	11	11	−	−	PROPN
ap-4589	11	12	d2	d2	PROPN
ap-4589	11	13	dx2	dx2	PROPN
ap-4589	11	14	+	+	CCONJ
ap-4589	11	15	j(j	j(j	PROPN
ap-4589	11	16	−	−	NOUN
ap-4589	12	1	1	1	X
ap-4589	12	2	)	)	PUNCT
ap-4589	12	3	x2	x2	NOUN
ap-4589	12	4	−	−	NOUN
ap-4589	12	5	2	2	NUM
ap-4589	12	6	g	g	NOUN
ap-4589	12	7	x	x	X
ap-4589	12	8	,	,	PUNCT
ap-4589	12	9	hspherical	hspherical	ADJ
ap-4589	12	10	kepler	kepler	NOUN
ap-4589	12	11	=	=	PUNCT
ap-4589	12	12	−	−	PROPN
ap-4589	12	13	d2	d2	PROPN
ap-4589	12	14	dx2	dx2	PROPN
ap-4589	12	15	+	+	CCONJ
ap-4589	12	16	j(j	j(j	PROPN
ap-4589	12	17	−	−	NOUN
ap-4589	12	18	1	1	NUM
ap-4589	12	19	)	)	PUNCT
ap-4589	12	20	sin2	sin2	NOUN
ap-4589	12	21	x	x	PUNCT
ap-4589	13	1	−	−	PROPN
ap-4589	13	2	2	2	NUM
ap-4589	13	3	g	g	NOUN
ap-4589	13	4	cotx	cotx	NOUN
ap-4589	13	5	,	,	PUNCT
ap-4589	13	6	hhyperbolic	hhyperbolic	ADJ
ap-4589	13	7	kepler	kepler	NOUN
ap-4589	13	8	=	=	PUNCT
ap-4589	13	9	−	−	PROPN
ap-4589	13	10	d2	d2	PROPN
ap-4589	13	11	dx2	dx2	PROPN
ap-4589	13	12	+	+	CCONJ
ap-4589	13	13	j(j	j(j	PROPN
ap-4589	13	14	−	−	NOUN
ap-4589	13	15	1	1	NUM
ap-4589	13	16	)	)	PUNCT
ap-4589	13	17	sinh2	sinh2	NOUN
ap-4589	13	18	x	x	PUNCT
ap-4589	14	1	−	−	PROPN
ap-4589	14	2	2	2	NUM
ap-4589	14	3	g	g	NOUN
ap-4589	14	4	coth	coth	NOUN
ap-4589	14	5	x	x	PROPN
ap-4589	14	6	,	,	PUNCT
ap-4589	14	7	hrosen	hrosen	NOUN
ap-4589	14	8	–	–	PUNCT
ap-4589	14	9	morse	morse	NOUN
ap-4589	14	10	=	=	SYM
ap-4589	14	11	−	−	PROPN
ap-4589	14	12	d2	d2	PROPN
ap-4589	14	13	dx2	dx2	PROPN
ap-4589	14	14	−	−	PROPN
ap-4589	14	15	j(j	j(j	PROPN
ap-4589	14	16	−	−	NOUN
ap-4589	14	17	1	1	NUM
ap-4589	14	18	)	)	PUNCT
ap-4589	14	19	cosh2	cosh2	VERB
ap-4589	15	1	x	x	PUNCT
ap-4589	15	2	−	−	PROPN
ap-4589	15	3	2	2	NUM
ap-4589	15	4	g	g	NOUN
ap-4589	15	5	tanh	tanh	PROPN
ap-4589	15	6	x	x	X
ap-4589	15	7	,	,	PUNCT
ap-4589	15	8	(	(	PUNCT
ap-4589	15	9	1.1	1.1	NUM
ap-4589	15	10	)	)	PUNCT
ap-4589	15	11	where	where	SCONJ
ap-4589	15	12	j	j	PROPN
ap-4589	15	13	and	and	CCONJ
ap-4589	15	14	g	g	PROPN
ap-4589	15	15	are	be	AUX
ap-4589	15	16	real	real	ADJ
ap-4589	15	17	parameters	parameter	NOUN
ap-4589	15	18	.	.	PUNCT
ap-4589	16	1	the	the	DET
ap-4589	16	2	potential	potential	ADJ
ap-4589	16	3	energies	energy	NOUN
ap-4589	16	4	and	and	CCONJ
ap-4589	16	5	bound	bind	VERB
ap-4589	16	6	-	-	PUNCT
ap-4589	16	7	state	state	NOUN
ap-4589	16	8	spectra	spectra	NOUN
ap-4589	16	9	are	be	AUX
ap-4589	16	10	depicted	depict	VERB
ap-4589	16	11	in	in	ADP
ap-4589	16	12	figure	figure	NOUN
ap-4589	16	13	1	1	NUM
ap-4589	16	14	.	.	PUNCT
ap-4589	17	1	there	there	PRON
ap-4589	17	2	exist	exist	VERB
ap-4589	17	3	several	several	ADJ
ap-4589	17	4	methods	method	NOUN
ap-4589	17	5	to	to	PART
ap-4589	17	6	solve	solve	VERB
ap-4589	17	7	the	the	DET
ap-4589	17	8	eigenvalue	eigenvalue	ADJ
ap-4589	17	9	problems	problem	NOUN
ap-4589	17	10	of	of	ADP
ap-4589	17	11	these	these	DET
ap-4589	17	12	hamiltonians	hamiltonian	NOUN
ap-4589	17	13	(	(	PUNCT
ap-4589	17	14	1.1	1.1	NUM
ap-4589	17	15	)	)	PUNCT
ap-4589	17	16	.	.	PUNCT
ap-4589	18	1	among	among	ADP
ap-4589	18	2	them	they	PRON
ap-4589	18	3	is	be	AUX
ap-4589	18	4	the	the	DET
ap-4589	18	5	shape	shape	NOUN
ap-4589	18	6	invariance	invariance	NOUN
ap-4589	18	7	method	method	NOUN
ap-4589	18	8	[	[	PUNCT
ap-4589	18	9	9],2	9],2	NUM
ap-4589	18	10	which	which	PRON
ap-4589	18	11	is	be	AUX
ap-4589	18	12	based	base	VERB
ap-4589	18	13	on	on	ADP
ap-4589	18	14	the	the	DET
ap-4589	18	15	factorization	factorization	NOUN
ap-4589	18	16	of	of	ADP
ap-4589	18	17	hamiltonian	hamiltonian	NOUN
ap-4589	18	18	and	and	CCONJ
ap-4589	18	19	the	the	DET
ap-4589	18	20	darboux	darboux	ADJ
ap-4589	18	21	transformation	transformation	NOUN
ap-4589	18	22	.	.	PUNCT
ap-4589	19	1	and	and	CCONJ
ap-4589	19	2	,	,	PUNCT
ap-4589	19	3	as	as	SCONJ
ap-4589	19	4	discussed	discuss	VERB
ap-4589	19	5	by	by	ADP
ap-4589	19	6	gangopadhyaya	gangopadhyaya	NOUN
ap-4589	19	7	et	et	PROPN
ap-4589	19	8	al	al	PROPN
ap-4589	19	9	.	.	PUNCT
ap-4589	20	1	[	[	X
ap-4589	20	2	11	11	NUM
ap-4589	20	3	]	]	PUNCT
ap-4589	20	4	(	(	PUNCT
ap-4589	20	5	see	see	VERB
ap-4589	20	6	also	also	ADV
ap-4589	20	7	the	the	DET
ap-4589	20	8	reviews	review	NOUN
ap-4589	20	9	[	[	X
ap-4589	20	10	12	12	NUM
ap-4589	20	11	,	,	PUNCT
ap-4589	20	12	13	13	NUM
ap-4589	20	13	]	]	NUM
ap-4589	20	14	)	)	PUNCT
ap-4589	20	15	,	,	PUNCT
ap-4589	20	16	the	the	DET
ap-4589	20	17	shape	shape	NOUN
ap-4589	20	18	invariance	invariance	NOUN
ap-4589	20	19	can	can	AUX
ap-4589	20	20	always	always	ADV
ap-4589	20	21	be	be	AUX
ap-4589	20	22	translated	translate	VERB
ap-4589	20	23	into	into	ADP
ap-4589	20	24	the	the	DET
ap-4589	20	25	(	(	PUNCT
ap-4589	20	26	lie-)algebraic	lie-)algebraic	ADJ
ap-4589	20	27	description	description	NOUN
ap-4589	20	28	—	—	PUNCT
ap-4589	20	29	the	the	DET
ap-4589	20	30	so	so	ADV
ap-4589	20	31	-	-	PUNCT
ap-4589	20	32	called	call	VERB
ap-4589	20	33	potential	potential	ADJ
ap-4589	20	34	algebra.3	algebra.3	PROPN
ap-4589	20	35	the	the	DET
ap-4589	20	36	spectral	spectral	ADJ
ap-4589	20	37	problem	problem	NOUN
ap-4589	20	38	can	can	AUX
ap-4589	20	39	then	then	ADV
ap-4589	20	40	be	be	AUX
ap-4589	20	41	solved	solve	VERB
ap-4589	20	42	by	by	ADP
ap-4589	20	43	means	mean	NOUN
ap-4589	20	44	of	of	ADP
ap-4589	20	45	representation	representation	NOUN
ap-4589	20	46	theory	theory	NOUN
ap-4589	20	47	.	.	PUNCT
ap-4589	21	1	however	however	ADV
ap-4589	21	2	,	,	PUNCT
ap-4589	21	3	as	as	ADV
ap-4589	21	4	far	far	ADV
ap-4589	21	5	as	as	SCONJ
ap-4589	21	6	we	we	PRON
ap-4589	21	7	noticed	notice	VERB
ap-4589	21	8	,	,	PUNCT
ap-4589	21	9	the	the	DET
ap-4589	21	10	representation	representation	NOUN
ap-4589	21	11	theory	theory	NOUN
ap-4589	21	12	of	of	ADP
ap-4589	21	13	potential	potential	ADJ
ap-4589	21	14	algebra	algebra	NOUN
ap-4589	21	15	has	have	AUX
ap-4589	21	16	not	not	PART
ap-4589	21	17	been	be	AUX
ap-4589	21	18	fully	fully	ADV
ap-4589	21	19	analyzed	analyze	VERB
ap-4589	21	20	yet	yet	ADV
ap-4589	21	21	.	.	PUNCT
ap-4589	22	1	in	in	ADP
ap-4589	22	2	particular	particular	ADJ
ap-4589	22	3	,	,	PUNCT
ap-4589	22	4	the	the	DET
ap-4589	22	5	spectral	spectral	ADJ
ap-4589	22	6	problems	problem	NOUN
ap-4589	22	7	of	of	ADP
ap-4589	22	8	the	the	DET
ap-4589	22	9	above	above	ADJ
ap-4589	22	10	hamiltonians	hamiltonian	NOUN
ap-4589	22	11	have	have	AUX
ap-4589	22	12	not	not	PART
ap-4589	22	13	been	be	AUX
ap-4589	22	14	solved	solve	VERB
ap-4589	22	15	in	in	ADP
ap-4589	22	16	terms	term	NOUN
ap-4589	22	17	of	of	ADP
ap-4589	22	18	potential	potential	ADJ
ap-4589	22	19	algebra	algebra	NOUN
ap-4589	22	20	.	.	PUNCT
ap-4589	23	1	the	the	DET
ap-4589	23	2	purpose	purpose	NOUN
ap-4589	23	3	of	of	ADP
ap-4589	23	4	this	this	DET
ap-4589	23	5	note	note	NOUN
ap-4589	23	6	is	be	AUX
ap-4589	23	7	to	to	PART
ap-4589	23	8	fill	fill	VERB
ap-4589	23	9	this	this	DET
ap-4589	23	10	gap	gap	NOUN
ap-4589	23	11	.	.	PUNCT
ap-4589	24	1	as	as	SCONJ
ap-4589	24	2	we	we	PRON
ap-4589	24	3	will	will	AUX
ap-4589	24	4	see	see	VERB
ap-4589	24	5	below	below	ADV
ap-4589	24	6	,	,	PUNCT
ap-4589	24	7	these	these	DET
ap-4589	24	8	very	very	ADV
ap-4589	24	9	old	old	ADJ
ap-4589	24	10	spectral	spectral	ADJ
ap-4589	24	11	problems	problem	NOUN
ap-4589	24	12	require	require	VERB
ap-4589	24	13	to	to	PART
ap-4589	24	14	introduce	introduce	VERB
ap-4589	24	15	rather	rather	ADV
ap-4589	24	16	nontrivial	nontrivial	ADJ
ap-4589	24	17	nonlinear	nonlinear	ADJ
ap-4589	24	18	algebraic	algebraic	PROPN
ap-4589	24	19	systems	system	NOUN
ap-4589	24	20	.	.	PUNCT
ap-4589	25	1	the	the	DET
ap-4589	25	2	goal	goal	NOUN
ap-4589	25	3	of	of	ADP
ap-4589	25	4	this	this	DET
ap-4589	25	5	note	note	NOUN
ap-4589	25	6	is	be	AUX
ap-4589	25	7	to	to	PART
ap-4589	25	8	show	show	VERB
ap-4589	25	9	that	that	SCONJ
ap-4589	25	10	these	these	DET
ap-4589	25	11	bound	bind	VERB
ap-4589	25	12	-	-	PUNCT
ap-4589	25	13	state	state	NOUN
ap-4589	25	14	problems	problem	NOUN
ap-4589	25	15	can	can	AUX
ap-4589	25	16	be	be	AUX
ap-4589	25	17	solved	solve	VERB
ap-4589	25	18	by	by	ADP
ap-4589	25	19	representation	representation	NOUN
ap-4589	25	20	theory	theory	NOUN
ap-4589	25	21	of	of	ADP
ap-4589	25	22	the	the	DET
ap-4589	25	23	operators	operator	NOUN
ap-4589	25	24	{	{	PUNCT
ap-4589	25	25	j3	j3	PROPN
ap-4589	25	26	,	,	PUNCT
ap-4589	25	27	j+	j+	NUM
ap-4589	25	28	,	,	PUNCT
ap-4589	25	29	j−	j−	PROPN
ap-4589	25	30	}	}	PUNCT
ap-4589	25	31	that	that	PRON
ap-4589	25	32	satisfy	satisfy	VERB
ap-4589	25	33	the	the	DET
ap-4589	25	34	linear	linear	ADJ
ap-4589	25	35	commutation	commutation	NOUN
ap-4589	25	36	relations	relation	NOUN
ap-4589	25	37	between	between	ADP
ap-4589	25	38	j3	j3	PROPN
ap-4589	25	39	and	and	CCONJ
ap-4589	25	40	j±	j±	PROPN
ap-4589	26	1	[	[	X
ap-4589	26	2	j3	j3	PROPN
ap-4589	26	3	,	,	PUNCT
ap-4589	26	4	j±	j±	PROPN
ap-4589	26	5	]	]	X
ap-4589	26	6	=	=	SYM
ap-4589	26	7	±j±	±j±	PROPN
ap-4589	26	8	,	,	PUNCT
ap-4589	26	9	(	(	PUNCT
ap-4589	26	10	1.2	1.2	NUM
ap-4589	26	11	)	)	PUNCT
ap-4589	26	12	and	and	CCONJ
ap-4589	26	13	the	the	DET
ap-4589	26	14	nonlinear	nonlinear	ADJ
ap-4589	26	15	commutation	commutation	NOUN
ap-4589	26	16	relations	relation	NOUN
ap-4589	26	17	between	between	ADP
ap-4589	26	18	j+	j+	PROPN
ap-4589	26	19	and	and	CCONJ
ap-4589	26	20	j−	j−	PROPN
ap-4589	26	21	(	(	PUNCT
ap-4589	26	22	kepler	kepler	PROPN
ap-4589	26	23	)	)	PUNCT
ap-4589	27	1	[	[	X
ap-4589	27	2	j+	j+	NUM
ap-4589	27	3	,	,	PUNCT
ap-4589	27	4	j−	j−	PROPN
ap-4589	27	5	]	]	X
ap-4589	27	6	=	=	PUNCT
ap-4589	28	1	−	−	PROPN
ap-4589	28	2	g	g	PROPN
ap-4589	28	3	2	2	NUM
ap-4589	28	4	j2	j2	NOUN
ap-4589	28	5	3	3	NUM
ap-4589	28	6	+	+	CCONJ
ap-4589	28	7	g2	g2	PROPN
ap-4589	28	8	(	(	PUNCT
ap-4589	28	9	j3	j3	PROPN
ap-4589	28	10	−	−	PROPN
ap-4589	28	11	1)2	1)2	PROPN
ap-4589	28	12	,	,	PUNCT
ap-4589	28	13	(	(	PUNCT
ap-4589	28	14	spherical	spherical	ADJ
ap-4589	28	15	kepler	kepler	NOUN
ap-4589	28	16	)	)	PUNCT
ap-4589	29	1	[	[	X
ap-4589	29	2	j+	j+	NUM
ap-4589	29	3	,	,	PUNCT
ap-4589	29	4	j−	j−	PROPN
ap-4589	29	5	]	]	X
ap-4589	29	6	=	=	SYM
ap-4589	29	7	j2	j2	PROPN
ap-4589	29	8	3	3	NUM
ap-4589	29	9	−	−	PROPN
ap-4589	29	10	g2	g2	PROPN
ap-4589	29	11	j2	j2	PROPN
ap-4589	29	12	3	3	NUM
ap-4589	29	13	−	−	PROPN
ap-4589	29	14	(	(	PUNCT
ap-4589	29	15	j3	j3	PROPN
ap-4589	29	16	−	−	PROPN
ap-4589	29	17	1)2	1)2	PROPN
ap-4589	29	18	+	+	CCONJ
ap-4589	29	19	g2	g2	PROPN
ap-4589	29	20	(	(	PUNCT
ap-4589	29	21	j3	j3	PROPN
ap-4589	29	22	−	−	PROPN
ap-4589	29	23	1)2	1)2	PROPN
ap-4589	29	24	,	,	PUNCT
ap-4589	29	25	(	(	PUNCT
ap-4589	29	26	hyperbolic	hyperbolic	ADJ
ap-4589	29	27	kepler	kepler	PROPN
ap-4589	29	28	&	&	CCONJ
ap-4589	29	29	rosen	rosen	PROPN
ap-4589	29	30	–	–	PUNCT
ap-4589	29	31	morse	morse	PROPN
ap-4589	29	32	)	)	PUNCT
ap-4589	30	1	[	[	X
ap-4589	30	2	j+	j+	NUM
ap-4589	30	3	,	,	PUNCT
ap-4589	30	4	j−	j−	PROPN
ap-4589	30	5	]	]	X
ap-4589	31	1	=	=	SYM
ap-4589	31	2	−j2	−j2	NOUN
ap-4589	31	3	3	3	NUM
ap-4589	31	4	−	−	PROPN
ap-4589	31	5	g2	g2	PROPN
ap-4589	31	6	j2	j2	PROPN
ap-4589	31	7	3	3	NUM
ap-4589	31	8	+	+	CCONJ
ap-4589	31	9	(	(	PUNCT
ap-4589	31	10	j3	j3	PROPN
ap-4589	31	11	−	−	PROPN
ap-4589	31	12	1)2	1)2	PROPN
ap-4589	31	13	+	+	CCONJ
ap-4589	31	14	g2	g2	PROPN
ap-4589	31	15	(	(	PUNCT
ap-4589	31	16	j3	j3	PROPN
ap-4589	31	17	−	−	PROPN
ap-4589	31	18	1)2	1)2	PROPN
ap-4589	31	19	.	.	PUNCT
ap-4589	32	1	(	(	PUNCT
ap-4589	32	2	1.3	1.3	NUM
ap-4589	32	3	)	)	PUNCT
ap-4589	32	4	we	we	PRON
ap-4589	32	5	will	will	AUX
ap-4589	32	6	see	see	VERB
ap-4589	32	7	that	that	SCONJ
ap-4589	32	8	,	,	PUNCT
ap-4589	32	9	if	if	SCONJ
ap-4589	32	10	j	j	PROPN
ap-4589	32	11	is	be	AUX
ap-4589	32	12	a	a	DET
ap-4589	32	13	half	half	ADJ
ap-4589	32	14	-	-	PUNCT
ap-4589	32	15	integer	integer	NOUN
ap-4589	32	16	,	,	PUNCT
ap-4589	32	17	the	the	DET
ap-4589	32	18	bound	bind	VERB
ap-4589	32	19	-	-	PUNCT
ap-4589	32	20	state	state	NOUN
ap-4589	32	21	problems	problem	NOUN
ap-4589	32	22	of	of	ADP
ap-4589	32	23	(	(	PUNCT
ap-4589	32	24	1.1	1.1	NUM
ap-4589	32	25	)	)	PUNCT
ap-4589	32	26	can	can	AUX
ap-4589	32	27	be	be	AUX
ap-4589	32	28	solved	solve	VERB
ap-4589	32	29	from	from	ADP
ap-4589	32	30	these	these	DET
ap-4589	32	31	operators	operator	NOUN
ap-4589	32	32	.	.	PUNCT
ap-4589	33	1	the	the	DET
ap-4589	33	2	rest	rest	NOUN
ap-4589	33	3	of	of	ADP
ap-4589	33	4	the	the	DET
ap-4589	33	5	note	note	NOUN
ap-4589	33	6	is	be	AUX
ap-4589	33	7	organized	organize	VERB
ap-4589	33	8	as	as	SCONJ
ap-4589	33	9	follows	follow	VERB
ap-4589	33	10	:	:	PUNCT
ap-4589	33	11	in	in	ADP
ap-4589	33	12	section	section	NOUN
ap-4589	33	13	2	2	NUM
ap-4589	33	14	we	we	PRON
ap-4589	33	15	introduce	introduce	VERB
ap-4589	33	16	the	the	DET
ap-4589	33	17	potential	potential	ADJ
ap-4589	33	18	algebra	algebra	NOUN
ap-4589	33	19	for	for	ADP
ap-4589	33	20	the	the	DET
ap-4589	33	21	kepler	kepler	PROPN
ap-4589	33	22	problem	problem	NOUN
ap-4589	33	23	in	in	ADP
ap-4589	33	24	flat	flat	ADJ
ap-4589	33	25	space	space	NOUN
ap-4589	33	26	and	and	CCONJ
ap-4589	33	27	solve	solve	VERB
ap-4589	33	28	the	the	DET
ap-4589	33	29	spectral	spectral	ADJ
ap-4589	33	30	problem	problem	NOUN
ap-4589	33	31	by	by	ADP
ap-4589	33	32	means	mean	NOUN
ap-4589	33	33	of	of	ADP
ap-4589	33	34	representation	representation	NOUN
ap-4589	33	35	theory	theory	NOUN
ap-4589	33	36	.	.	PUNCT
ap-4589	34	1	in	in	ADP
ap-4589	34	2	sections	section	NOUN
ap-4589	34	3	3	3	NUM
ap-4589	34	4	and	and	CCONJ
ap-4589	34	5	4	4	NUM
ap-4589	34	6	1these	1these	NUM
ap-4589	34	7	names	name	NOUN
ap-4589	34	8	for	for	ADP
ap-4589	34	9	the	the	DET
ap-4589	34	10	hamiltonians	hamiltonian	NOUN
ap-4589	34	11	,	,	PUNCT
ap-4589	34	12	though	though	SCONJ
ap-4589	34	13	not	not	PART
ap-4589	34	14	so	so	ADV
ap-4589	34	15	popular	popular	ADJ
ap-4589	34	16	nowadays	nowadays	ADV
ap-4589	34	17	,	,	PUNCT
ap-4589	34	18	are	be	AUX
ap-4589	34	19	borrowed	borrow	VERB
ap-4589	34	20	(	(	PUNCT
ap-4589	34	21	with	with	ADP
ap-4589	34	22	slight	slight	ADJ
ap-4589	34	23	modifications	modification	NOUN
ap-4589	34	24	)	)	PUNCT
ap-4589	34	25	from	from	ADP
ap-4589	34	26	infeld	infeld	NOUN
ap-4589	34	27	and	and	CCONJ
ap-4589	34	28	hull	hull	NOUN
ap-4589	34	29	[	[	X
ap-4589	34	30	8	8	NUM
ap-4589	34	31	]	]	PUNCT
ap-4589	34	32	.	.	PUNCT
ap-4589	35	1	notice	notice	VERB
ap-4589	35	2	that	that	SCONJ
ap-4589	35	3	these	these	PRON
ap-4589	35	4	are	be	AUX
ap-4589	35	5	different	different	ADJ
ap-4589	35	6	from	from	ADP
ap-4589	35	7	those	those	PRON
ap-4589	35	8	commonly	commonly	ADV
ap-4589	35	9	used	use	VERB
ap-4589	35	10	in	in	ADP
ap-4589	35	11	the	the	DET
ap-4589	35	12	supersymmetric	supersymmetric	ADJ
ap-4589	35	13	quantum	quantum	ADJ
ap-4589	35	14	mechanics	mechanic	NOUN
ap-4589	35	15	literature	literature	NOUN
ap-4589	36	1	[	[	X
ap-4589	36	2	9	9	NUM
ap-4589	36	3	]	]	PUNCT
ap-4589	36	4	.	.	PUNCT
ap-4589	37	1	2recently	2recently	ADV
ap-4589	37	2	it	it	PRON
ap-4589	37	3	has	have	AUX
ap-4589	37	4	been	be	AUX
ap-4589	37	5	demonstrated	demonstrate	VERB
ap-4589	37	6	that	that	SCONJ
ap-4589	37	7	spectral	spectral	ADJ
ap-4589	37	8	intertwining	intertwine	VERB
ap-4589	37	9	relation	relation	NOUN
ap-4589	37	10	provides	provide	VERB
ap-4589	37	11	a	a	DET
ap-4589	37	12	yet	yet	ADV
ap-4589	37	13	another	another	DET
ap-4589	37	14	scheme	scheme	NOUN
ap-4589	37	15	to	to	PART
ap-4589	37	16	solve	solve	VERB
ap-4589	37	17	the	the	DET
ap-4589	37	18	eigenvalue	eigenvalue	ADJ
ap-4589	37	19	problems	problem	NOUN
ap-4589	37	20	of	of	ADP
ap-4589	37	21	hkepler	hkepler	NOUN
ap-4589	37	22	,	,	PUNCT
ap-4589	37	23	hspherical	hspherical	ADJ
ap-4589	37	24	kepler	kepler	NOUN
ap-4589	37	25	,	,	PUNCT
ap-4589	37	26	and	and	CCONJ
ap-4589	37	27	hhyperbolic	hhyperbolic	ADJ
ap-4589	37	28	kepler	kepler	NOUN
ap-4589	38	1	[	[	X
ap-4589	38	2	10	10	NUM
ap-4589	38	3	]	]	PUNCT
ap-4589	38	4	.	.	PUNCT
ap-4589	39	1	3a	3a	PROPN
ap-4589	39	2	similar	similar	ADJ
ap-4589	39	3	algebraic	algebraic	ADJ
ap-4589	39	4	description	description	NOUN
ap-4589	39	5	for	for	ADP
ap-4589	39	6	shape	shape	NOUN
ap-4589	39	7	invariance	invariance	NOUN
ap-4589	39	8	has	have	AUX
ap-4589	39	9	also	also	ADV
ap-4589	39	10	been	be	AUX
ap-4589	39	11	discussed	discuss	VERB
ap-4589	39	12	by	by	ADP
ap-4589	39	13	balantekin	balantekin	NOUN
ap-4589	39	14	[	[	X
ap-4589	39	15	14	14	NUM
ap-4589	39	16	]	]	SYM
ap-4589	39	17	.	.	PUNCT
ap-4589	40	1	446	446	NUM
ap-4589	40	2	http://dx.doi.org/10.14311/ap.2017.57.0446	http://dx.doi.org/10.14311/ap.2017.57.0446	ADP
ap-4589	40	3	http://ojs.cvut.cz/ojs/index.php/ap	http://ojs.cvut.cz/ojs/index.php/ap	PROPN
ap-4589	40	4	vol	vol	NOUN
ap-4589	40	5	.	.	PUNCT
ap-4589	41	1	57	57	NUM
ap-4589	41	2	no	no	NOUN
ap-4589	41	3	.	.	PUNCT
ap-4589	42	1	6/2017	6/2017	X
ap-4589	42	2	algebraic	algebraic	ADJ
ap-4589	42	3	description	description	NOUN
ap-4589	42	4	of	of	ADP
ap-4589	42	5	shape	shape	NOUN
ap-4589	42	6	invariance	invariance	NOUN
ap-4589	42	7	revisited	revisit	VERB
ap-4589	42	8	0	0	NUM
ap-4589	42	9	x	x	SYM
ap-4589	42	10	v	v	NOUN
ap-4589	42	11	(	(	PUNCT
ap-4589	42	12	x	x	X
ap-4589	42	13	)	)	PUNCT
ap-4589	42	14	(	(	PUNCT
ap-4589	42	15	a	a	X
ap-4589	42	16	)	)	PUNCT
ap-4589	42	17	kepler	kepler	NOUN
ap-4589	42	18	0	0	NUM
ap-4589	43	1	π	π	NOUN
ap-4589	43	2	v	v	X
ap-4589	43	3	(	(	PUNCT
ap-4589	43	4	x	x	X
ap-4589	43	5	)	)	PUNCT
ap-4589	43	6	(	(	PUNCT
ap-4589	43	7	b	b	X
ap-4589	43	8	)	)	PUNCT
ap-4589	43	9	spherical	spherical	ADJ
ap-4589	43	10	kepler	kepler	NOUN
ap-4589	43	11	0	0	NUM
ap-4589	43	12	x	x	SYM
ap-4589	43	13	v	v	NOUN
ap-4589	43	14	(	(	PUNCT
ap-4589	43	15	x	x	X
ap-4589	43	16	)	)	PUNCT
ap-4589	43	17	(	(	PUNCT
ap-4589	43	18	c	c	X
ap-4589	43	19	)	)	PUNCT
ap-4589	43	20	hyperbolic	hyperbolic	ADJ
ap-4589	43	21	kepler	kepler	NOUN
ap-4589	43	22	0	0	NUM
ap-4589	43	23	x	x	SYM
ap-4589	43	24	v	v	NOUN
ap-4589	43	25	(	(	PUNCT
ap-4589	43	26	x	x	NOUN
ap-4589	43	27	)	)	PUNCT
ap-4589	43	28	(	(	PUNCT
ap-4589	43	29	d	d	X
ap-4589	43	30	)	)	PUNCT
ap-4589	43	31	rosen	rosen	PROPN
ap-4589	43	32	–	–	PUNCT
ap-4589	43	33	morse	morse	ADJ
ap-4589	43	34	figure	figure	NOUN
ap-4589	43	35	1	1	NUM
ap-4589	43	36	.	.	PUNCT
ap-4589	43	37	potential	potential	ADJ
ap-4589	43	38	energies	energy	NOUN
ap-4589	43	39	(	(	PUNCT
ap-4589	43	40	thick	thick	ADJ
ap-4589	43	41	solid	solid	ADJ
ap-4589	43	42	curves	curve	NOUN
ap-4589	43	43	)	)	PUNCT
ap-4589	43	44	and	and	CCONJ
ap-4589	43	45	discrete	discrete	ADJ
ap-4589	43	46	energy	energy	NOUN
ap-4589	43	47	levels	level	NOUN
ap-4589	43	48	(	(	PUNCT
ap-4589	43	49	blue	blue	ADJ
ap-4589	43	50	lines	line	NOUN
ap-4589	43	51	)	)	PUNCT
ap-4589	43	52	.	.	PUNCT
ap-4589	44	1	we	we	PRON
ap-4589	44	2	generalize	generalize	VERB
ap-4589	44	3	to	to	ADP
ap-4589	44	4	the	the	DET
ap-4589	44	5	other	other	ADJ
ap-4589	44	6	problems	problem	NOUN
ap-4589	44	7	.	.	PUNCT
ap-4589	45	1	we	we	PRON
ap-4589	45	2	shall	shall	AUX
ap-4589	45	3	see	see	VERB
ap-4589	45	4	that	that	SCONJ
ap-4589	45	5	the	the	DET
ap-4589	45	6	bound	bind	VERB
ap-4589	45	7	-	-	PUNCT
ap-4589	45	8	state	state	NOUN
ap-4589	45	9	spectra	spectra	NOUN
ap-4589	45	10	of	of	ADP
ap-4589	45	11	the	the	DET
ap-4589	45	12	hyperbolic	hyperbolic	ADJ
ap-4589	45	13	kepler	kepler	PROPN
ap-4589	45	14	and	and	CCONJ
ap-4589	45	15	rosen	rosen	PROPN
ap-4589	45	16	–	–	PUNCT
ap-4589	45	17	morse	morse	PROPN
ap-4589	45	18	hamiltonians	hamiltonian	NOUN
ap-4589	45	19	just	just	ADV
ap-4589	45	20	correspond	correspond	VERB
ap-4589	45	21	to	to	ADP
ap-4589	45	22	two	two	NUM
ap-4589	45	23	distinct	distinct	ADJ
ap-4589	45	24	representations	representation	NOUN
ap-4589	45	25	of	of	ADP
ap-4589	45	26	the	the	DET
ap-4589	45	27	same	same	ADJ
ap-4589	45	28	algebraic	algebraic	ADJ
ap-4589	45	29	system	system	NOUN
ap-4589	45	30	.	.	PUNCT
ap-4589	46	1	we	we	PRON
ap-4589	46	2	conclude	conclude	VERB
ap-4589	46	3	in	in	ADP
ap-4589	46	4	section	section	NOUN
ap-4589	46	5	5	5	NUM
ap-4589	46	6	.	.	NOUN
ap-4589	46	7	2	2	NUM
ap-4589	46	8	.	.	X
ap-4589	47	1	kepler	kepler	PROPN
ap-4589	47	2	let	let	VERB
ap-4589	47	3	us	we	PRON
ap-4589	47	4	start	start	VERB
ap-4589	47	5	with	with	ADP
ap-4589	47	6	the	the	DET
ap-4589	47	7	kepler	kepler	PROPN
ap-4589	47	8	problem	problem	NOUN
ap-4589	47	9	in	in	ADP
ap-4589	47	10	flat	flat	ADJ
ap-4589	47	11	space	space	NOUN
ap-4589	47	12	.	.	PUNCT
ap-4589	48	1	as	as	SCONJ
ap-4589	48	2	is	be	AUX
ap-4589	48	3	well	well	ADV
ap-4589	48	4	known	know	VERB
ap-4589	48	5	,	,	PUNCT
ap-4589	48	6	the	the	DET
ap-4589	48	7	kepler	kepler	PROPN
ap-4589	48	8	hamiltonian	hamiltonian	PROPN
ap-4589	48	9	hkepler	hkepler	NOUN
ap-4589	48	10	in	in	ADP
ap-4589	48	11	(	(	PUNCT
ap-4589	48	12	1.1	1.1	NUM
ap-4589	48	13	)	)	PUNCT
ap-4589	48	14	can	can	AUX
ap-4589	48	15	be	be	AUX
ap-4589	48	16	factorized	factorize	VERB
ap-4589	48	17	as	as	SCONJ
ap-4589	48	18	follows	follow	VERB
ap-4589	48	19	:	:	PUNCT
ap-4589	48	20	hkepler	hkepler	NOUN
ap-4589	48	21	=	=	SYM
ap-4589	48	22	a−(j)a+(j)−	a−(j)a+(j)−	PROPN
ap-4589	48	23	g2	g2	PROPN
ap-4589	48	24	j2	j2	PROPN
ap-4589	48	25	,	,	PUNCT
ap-4589	48	26	(	(	PUNCT
ap-4589	48	27	2.1	2.1	NUM
ap-4589	48	28	)	)	PUNCT
ap-4589	48	29	where	where	SCONJ
ap-4589	48	30	a±(j	a±(j	PROPN
ap-4589	48	31	)	)	PUNCT
ap-4589	48	32	are	be	AUX
ap-4589	48	33	the	the	DET
ap-4589	48	34	first	first	ADJ
ap-4589	48	35	-	-	PUNCT
ap-4589	48	36	order	order	NOUN
ap-4589	48	37	differential	differential	NOUN
ap-4589	48	38	operators	operator	NOUN
ap-4589	48	39	given	give	VERB
ap-4589	48	40	by	by	ADP
ap-4589	48	41	a±(j	a±(j	PROPN
ap-4589	48	42	)	)	PUNCT
ap-4589	48	43	=	=	SYM
ap-4589	48	44	±	±	NUM
ap-4589	48	45	d	d	NOUN
ap-4589	48	46	dx	dx	PROPN
ap-4589	49	1	−	−	PROPN
ap-4589	49	2	j	j	PROPN
ap-4589	49	3	x	x	X
ap-4589	50	1	+	+	CCONJ
ap-4589	50	2	g	g	PROPN
ap-4589	50	3	j	j	PROPN
ap-4589	50	4	.	.	PUNCT
ap-4589	51	1	(	(	PUNCT
ap-4589	51	2	2.2	2.2	NUM
ap-4589	51	3	)	)	PUNCT
ap-4589	51	4	let	let	VERB
ap-4589	51	5	us	we	PRON
ap-4589	51	6	next	next	ADJ
ap-4589	51	7	introduce	introduce	VERB
ap-4589	51	8	the	the	DET
ap-4589	51	9	potential	potential	ADJ
ap-4589	51	10	algebra	algebra	NOUN
ap-4589	51	11	of	of	ADP
ap-4589	51	12	this	this	DET
ap-4589	51	13	system	system	NOUN
ap-4589	51	14	.	.	PUNCT
ap-4589	52	1	following	follow	VERB
ap-4589	52	2	[	[	X
ap-4589	52	3	11	11	NUM
ap-4589	52	4	]	]	PUNCT
ap-4589	52	5	with	with	ADP
ap-4589	52	6	slight	slight	ADJ
ap-4589	52	7	modifications	modification	NOUN
ap-4589	52	8	,	,	PUNCT
ap-4589	52	9	we	we	PRON
ap-4589	52	10	first	first	ADV
ap-4589	52	11	introduce	introduce	VERB
ap-4589	52	12	an	an	DET
ap-4589	52	13	auxiliary	auxiliary	ADJ
ap-4589	52	14	periodic	periodic	ADJ
ap-4589	52	15	variable	variable	NOUN
ap-4589	52	16	θ	θ	PROPN
ap-4589	52	17	∈	∈	PROPN
ap-4589	53	1	[	[	X
ap-4589	53	2	0	0	NUM
ap-4589	53	3	,	,	PUNCT
ap-4589	53	4	2π	2π	NOUN
ap-4589	53	5	)	)	PUNCT
ap-4589	53	6	,	,	PUNCT
ap-4589	53	7	then	then	ADV
ap-4589	53	8	upgrade	upgrade	VERB
ap-4589	53	9	the	the	DET
ap-4589	53	10	parameter	parameter	NOUN
ap-4589	53	11	j	j	PROPN
ap-4589	53	12	to	to	ADP
ap-4589	53	13	an	an	DET
ap-4589	53	14	operator	operator	NOUN
ap-4589	53	15	j3	j3	PROPN
ap-4589	53	16	=	=	PUNCT
ap-4589	53	17	−i∂θ	−i∂θ	PROPN
ap-4589	53	18	,	,	PUNCT
ap-4589	53	19	and	and	CCONJ
ap-4589	53	20	then	then	ADV
ap-4589	53	21	replace	replace	VERB
ap-4589	53	22	a+(j	a+(j	NOUN
ap-4589	53	23	)	)	PUNCT
ap-4589	53	24	and	and	CCONJ
ap-4589	53	25	a−(j	a−(j	PROPN
ap-4589	53	26	)	)	PUNCT
ap-4589	53	27	to	to	ADP
ap-4589	53	28	j+	j+	PROPN
ap-4589	53	29	=	=	SYM
ap-4589	53	30	eiθa+(j3	eiθa+(j3	NOUN
ap-4589	53	31	)	)	PUNCT
ap-4589	53	32	and	and	CCONJ
ap-4589	53	33	j−	j−	PROPN
ap-4589	53	34	=	=	SYM
ap-4589	53	35	a−(j3)e−iθ	a−(j3)e−iθ	PROPN
ap-4589	53	36	.	.	PUNCT
ap-4589	54	1	the	the	DET
ap-4589	54	2	resultant	resultant	NOUN
ap-4589	54	3	operators	operator	NOUN
ap-4589	54	4	that	that	PRON
ap-4589	54	5	we	we	PRON
ap-4589	54	6	wish	wish	VERB
ap-4589	54	7	to	to	PART
ap-4589	54	8	study	study	VERB
ap-4589	54	9	are	be	AUX
ap-4589	54	10	thus	thus	ADV
ap-4589	54	11	as	as	SCONJ
ap-4589	54	12	follows	follow	VERB
ap-4589	54	13	:	:	PUNCT
ap-4589	54	14	j3	j3	PROPN
ap-4589	54	15	=	=	PUNCT
ap-4589	54	16	−i∂θ	−i∂θ	PROPN
ap-4589	54	17	,	,	PUNCT
ap-4589	54	18	j+	j+	NUM
ap-4589	54	19	=	=	SYM
ap-4589	54	20	eiθ	eiθ	PROPN
ap-4589	54	21	(	(	PUNCT
ap-4589	54	22	∂x	∂x	PROPN
ap-4589	54	23	−	−	PROPN
ap-4589	54	24	j3	j3	PROPN
ap-4589	54	25	x	x	PUNCT
ap-4589	55	1	+	+	CCONJ
ap-4589	55	2	g	g	PROPN
ap-4589	55	3	j3	j3	PROPN
ap-4589	55	4	)	)	PUNCT
ap-4589	55	5	,	,	PUNCT
ap-4589	55	6	j−	j−	PROPN
ap-4589	55	7	=	=	PUNCT
ap-4589	55	8	(	(	PUNCT
ap-4589	55	9	−∂x	−∂x	NUM
ap-4589	55	10	−	−	PROPN
ap-4589	55	11	j3	j3	PROPN
ap-4589	55	12	x	x	PUNCT
ap-4589	56	1	+	+	CCONJ
ap-4589	56	2	g	g	PROPN
ap-4589	56	3	j3	j3	PROPN
ap-4589	56	4	)	)	PUNCT
ap-4589	56	5	e−iθ	e−iθ	PROPN
ap-4589	56	6	.	.	PUNCT
ap-4589	57	1	(	(	PUNCT
ap-4589	57	2	2.3	2.3	NUM
ap-4589	57	3	)	)	PUNCT
ap-4589	57	4	here	here	ADV
ap-4589	57	5	one	one	PRON
ap-4589	57	6	may	may	AUX
ap-4589	57	7	wonder	wonder	VERB
ap-4589	57	8	about	about	ADP
ap-4589	57	9	the	the	DET
ap-4589	57	10	meaning	meaning	NOUN
ap-4589	57	11	of	of	ADP
ap-4589	57	12	1	1	NUM
ap-4589	57	13	/	/	SYM
ap-4589	57	14	j3	j3	PROPN
ap-4589	57	15	.	.	PUNCT
ap-4589	58	1	the	the	DET
ap-4589	58	2	operator	operator	NOUN
ap-4589	58	3	1	1	NUM
ap-4589	58	4	/	/	SYM
ap-4589	58	5	j3	j3	PROPN
ap-4589	58	6	would	would	AUX
ap-4589	58	7	be	be	AUX
ap-4589	58	8	defined	define	VERB
ap-4589	58	9	as	as	ADP
ap-4589	58	10	the	the	DET
ap-4589	58	11	spectral	spectral	ADJ
ap-4589	58	12	decomposition	decomposition	NOUN
ap-4589	58	13	1	1	NUM
ap-4589	58	14	/	/	SYM
ap-4589	58	15	j3	j3	PROPN
ap-4589	58	16	=	=	SYM
ap-4589	58	17	∑	∑	PUNCT
ap-4589	58	18	j(1	j(1	PROPN
ap-4589	58	19	/	/	SYM
ap-4589	58	20	j)pj	j)pj	PROPN
ap-4589	58	21	,	,	PUNCT
ap-4589	58	22	where	where	SCONJ
ap-4589	58	23	pj	pj	PROPN
ap-4589	58	24	stands	stand	VERB
ap-4589	58	25	for	for	ADP
ap-4589	58	26	the	the	DET
ap-4589	58	27	projection	projection	NOUN
ap-4589	58	28	operator	operator	NOUN
ap-4589	58	29	onto	onto	ADP
ap-4589	58	30	the	the	DET
ap-4589	58	31	eigenspace	eigenspace	NOUN
ap-4589	58	32	of	of	ADP
ap-4589	58	33	j3	j3	PROPN
ap-4589	58	34	with	with	ADP
ap-4589	58	35	eigenvalue	eigenvalue	PROPN
ap-4589	58	36	j.	j.	PROPN
ap-4589	58	37	this	this	DET
ap-4589	58	38	definition	definition	NOUN
ap-4589	58	39	would	would	AUX
ap-4589	58	40	be	be	AUX
ap-4589	58	41	well	well	ADV
ap-4589	58	42	-	-	PUNCT
ap-4589	58	43	defined	define	VERB
ap-4589	58	44	unless	unless	SCONJ
ap-4589	58	45	the	the	DET
ap-4589	58	46	spectrum	spectrum	NOUN
ap-4589	58	47	of	of	ADP
ap-4589	58	48	j3	j3	PROPN
ap-4589	58	49	contains	contain	VERB
ap-4589	58	50	j	j	PROPN
ap-4589	59	1	=	=	PUNCT
ap-4589	59	2	0	0	PROPN
ap-4589	59	3	.	.	PUNCT
ap-4589	60	1	an	an	DET
ap-4589	60	2	alternative	alternative	ADJ
ap-4589	60	3	way	way	NOUN
ap-4589	60	4	to	to	PART
ap-4589	60	5	give	give	VERB
ap-4589	60	6	a	a	DET
ap-4589	60	7	meaning	meaning	NOUN
ap-4589	60	8	to	to	ADP
ap-4589	60	9	1	1	NUM
ap-4589	60	10	/	/	SYM
ap-4589	60	11	j3	j3	PROPN
ap-4589	60	12	would	would	AUX
ap-4589	60	13	be	be	AUX
ap-4589	60	14	the	the	DET
ap-4589	60	15	(	(	PUNCT
ap-4589	60	16	formal	formal	ADJ
ap-4589	60	17	)	)	PUNCT
ap-4589	60	18	power	power	NOUN
ap-4589	60	19	series	series	NOUN
ap-4589	60	20	1	1	NUM
ap-4589	60	21	j3	j3	PROPN
ap-4589	60	22	=	=	SYM
ap-4589	60	23	1	1	NUM
ap-4589	60	24	λ	λ	NOUN
ap-4589	60	25	1	1	NUM
ap-4589	60	26	1−(1−j3	1−(1−j3	NUM
ap-4589	60	27	/	/	SYM
ap-4589	60	28	λ	λ	NOUN
ap-4589	60	29	)	)	PUNCT
ap-4589	60	30	=	=	SYM
ap-4589	60	31	1	1	NUM
ap-4589	60	32	λ	λ	PROPN
ap-4589	60	33	∑∞	∑∞	NOUN
ap-4589	60	34	n=0(1	n=0(1	PROPN
ap-4589	60	35	−	−	PROPN
ap-4589	60	36	j3	j3	PROPN
ap-4589	60	37	λ	λ	PROPN
ap-4589	60	38	)	)	PUNCT
ap-4589	60	39	n	n	CCONJ
ap-4589	60	40	,	,	PUNCT
ap-4589	60	41	where	where	SCONJ
ap-4589	60	42	447	447	NUM
ap-4589	60	43	satoshi	satoshi	PROPN
ap-4589	60	44	ohya	ohya	PROPN
ap-4589	60	45	acta	acta	PROPN
ap-4589	60	46	polytechnica	polytechnica	PROPN
ap-4589	60	47	λ	λ	PROPN
ap-4589	60	48	is	be	AUX
ap-4589	60	49	an	an	DET
ap-4589	60	50	arbitrary	arbitrary	ADJ
ap-4589	60	51	constant	constant	ADJ
ap-4589	60	52	.	.	PUNCT
ap-4589	61	1	this	this	DET
ap-4589	61	2	expression	expression	NOUN
ap-4589	61	3	would	would	AUX
ap-4589	61	4	be	be	AUX
ap-4589	61	5	well	well	ADV
ap-4589	61	6	-	-	PUNCT
ap-4589	61	7	defined	define	VERB
ap-4589	61	8	if	if	SCONJ
ap-4589	61	9	the	the	DET
ap-4589	61	10	operator	operator	NOUN
ap-4589	61	11	norm	norm	NOUN
ap-4589	61	12	of	of	ADP
ap-4589	61	13	1	1	NUM
ap-4589	61	14	−	−	PROPN
ap-4589	61	15	j3	j3	PROPN
ap-4589	61	16	/	/	SYM
ap-4589	61	17	λ	λ	PROPN
ap-4589	61	18	satisfies	satisfie	NOUN
ap-4589	61	19	‖1−	‖1−	PROPN
ap-4589	61	20	j3	j3	PROPN
ap-4589	61	21	/	/	SYM
ap-4589	61	22	λ‖	λ‖	PROPN
ap-4589	61	23	<	<	X
ap-4589	61	24	1	1	X
ap-4589	61	25	.	.	X
ap-4589	61	26	for	for	ADP
ap-4589	61	27	the	the	DET
ap-4589	61	28	moment	moment	NOUN
ap-4589	61	29	,	,	PUNCT
ap-4589	61	30	however	however	ADV
ap-4589	61	31	,	,	PUNCT
ap-4589	61	32	we	we	PRON
ap-4589	61	33	will	will	AUX
ap-4589	61	34	proceed	proceed	VERB
ap-4589	61	35	the	the	DET
ap-4589	61	36	discussion	discussion	NOUN
ap-4589	61	37	at	at	ADP
ap-4589	61	38	the	the	DET
ap-4589	61	39	formal	formal	ADJ
ap-4589	61	40	level	level	NOUN
ap-4589	61	41	.	.	PUNCT
ap-4589	62	1	it	it	PRON
ap-4589	62	2	is	be	AUX
ap-4589	62	3	not	not	PART
ap-4589	62	4	difficult	difficult	ADJ
ap-4589	62	5	to	to	PART
ap-4589	62	6	show	show	VERB
ap-4589	62	7	that	that	SCONJ
ap-4589	62	8	the	the	DET
ap-4589	62	9	operators	operator	NOUN
ap-4589	62	10	(	(	PUNCT
ap-4589	62	11	2.3	2.3	NUM
ap-4589	62	12	)	)	PUNCT
ap-4589	62	13	satisfy	satisfy	VERB
ap-4589	62	14	the	the	DET
ap-4589	62	15	following	follow	VERB
ap-4589	62	16	commutation	commutation	NOUN
ap-4589	62	17	relations	relation	NOUN
ap-4589	62	18	:	:	PUNCT
ap-4589	63	1	[	[	X
ap-4589	63	2	j3	j3	PROPN
ap-4589	63	3	,	,	PUNCT
ap-4589	63	4	j±	j±	PROPN
ap-4589	63	5	]	]	X
ap-4589	63	6	=	=	SYM
ap-4589	63	7	±j±	±j±	PROPN
ap-4589	63	8	,	,	PUNCT
ap-4589	63	9	[	[	X
ap-4589	63	10	j+	j+	NUM
ap-4589	63	11	,	,	PUNCT
ap-4589	63	12	j−	j−	PROPN
ap-4589	63	13	]	]	X
ap-4589	63	14	=	=	PUNCT
ap-4589	63	15	−	−	PROPN
ap-4589	63	16	g	g	PROPN
ap-4589	63	17	2	2	NUM
ap-4589	63	18	j2	j2	NOUN
ap-4589	63	19	3	3	NUM
ap-4589	63	20	+	+	CCONJ
ap-4589	63	21	g2	g2	PROPN
ap-4589	63	22	(	(	PUNCT
ap-4589	63	23	j3	j3	PROPN
ap-4589	63	24	−	−	PROPN
ap-4589	63	25	1)2	1)2	NUM
ap-4589	63	26	,	,	PUNCT
ap-4589	63	27	(	(	PUNCT
ap-4589	63	28	2.4	2.4	NUM
ap-4589	63	29	)	)	PUNCT
ap-4589	63	30	which	which	PRON
ap-4589	63	31	follow	follow	VERB
ap-4589	63	32	from	from	ADP
ap-4589	63	33	e∓iθj3e±iθ	e∓iθj3e±iθ	PROPN
ap-4589	63	34	=	=	SYM
ap-4589	63	35	j3	j3	PROPN
ap-4589	63	36	±	±	PROPN
ap-4589	63	37	1	1	NUM
ap-4589	63	38	or	or	CCONJ
ap-4589	63	39	j3e±iθ	j3e±iθ	NOUN
ap-4589	63	40	=	=	PUNCT
ap-4589	63	41	e±iθ(j3	e±iθ(j3	NUM
ap-4589	63	42	±	±	NUM
ap-4589	63	43	1	1	NUM
ap-4589	63	44	)	)	PUNCT
ap-4589	63	45	.	.	PUNCT
ap-4589	64	1	it	it	PRON
ap-4589	64	2	is	be	AUX
ap-4589	64	3	also	also	ADV
ap-4589	64	4	easy	easy	ADJ
ap-4589	64	5	to	to	PART
ap-4589	64	6	check	check	VERB
ap-4589	64	7	that	that	SCONJ
ap-4589	64	8	the	the	DET
ap-4589	64	9	invariant	invariant	ADJ
ap-4589	64	10	operator	operator	NOUN
ap-4589	64	11	of	of	ADP
ap-4589	64	12	this	this	DET
ap-4589	64	13	algebraic	algebraic	ADJ
ap-4589	64	14	system	system	NOUN
ap-4589	64	15	is	be	AUX
ap-4589	64	16	given	give	VERB
ap-4589	64	17	by	by	ADP
ap-4589	64	18	h	h	NOUN
ap-4589	64	19	=	=	SYM
ap-4589	64	20	j−j+	j−j+	PROPN
ap-4589	64	21	−	−	PROPN
ap-4589	64	22	g2	g2	PROPN
ap-4589	64	23	j2	j2	PROPN
ap-4589	64	24	3	3	NUM
ap-4589	64	25	=	=	SYM
ap-4589	64	26	j+j−	j+j−	PROPN
ap-4589	64	27	−	−	PROPN
ap-4589	64	28	g2	g2	PROPN
ap-4589	64	29	(	(	PUNCT
ap-4589	64	30	j3	j3	PROPN
ap-4589	64	31	−	−	PROPN
ap-4589	65	1	1)2	1)2	NUM
ap-4589	66	1	=	=	PUNCT
ap-4589	66	2	−∂2	−∂2	NOUN
ap-4589	66	3	x	x	PUNCT
ap-4589	66	4	+	+	PUNCT
ap-4589	66	5	j3(j3	j3(j3	NUM
ap-4589	66	6	−	−	PROPN
ap-4589	66	7	1	1	X
ap-4589	66	8	)	)	PUNCT
ap-4589	66	9	x2	x2	NOUN
ap-4589	66	10	−	−	NOUN
ap-4589	66	11	2	2	NUM
ap-4589	66	12	g	g	NOUN
ap-4589	66	13	x	x	SYM
ap-4589	66	14	,	,	PUNCT
ap-4589	66	15	(	(	PUNCT
ap-4589	66	16	2.5	2.5	NUM
ap-4589	66	17	)	)	PUNCT
ap-4589	66	18	which	which	PRON
ap-4589	66	19	commutes	commute	VERB
ap-4589	66	20	with	with	ADP
ap-4589	66	21	j±	j±	PROPN
ap-4589	66	22	and	and	CCONJ
ap-4589	66	23	j3.4	j3.4	VERB
ap-4589	66	24	notice	notice	VERB
ap-4589	66	25	that	that	SCONJ
ap-4589	66	26	if	if	SCONJ
ap-4589	66	27	g	g	PROPN
ap-4589	66	28	=	=	NOUN
ap-4589	66	29	0	0	PROPN
ap-4589	66	30	the	the	DET
ap-4589	66	31	commutation	commutation	NOUN
ap-4589	66	32	relations	relation	NOUN
ap-4589	66	33	(	(	PUNCT
ap-4589	66	34	2.4	2.4	NUM
ap-4589	66	35	)	)	PUNCT
ap-4589	66	36	just	just	ADV
ap-4589	66	37	describe	describe	VERB
ap-4589	66	38	those	those	PRON
ap-4589	66	39	for	for	ADP
ap-4589	66	40	the	the	DET
ap-4589	66	41	lie	lie	NOUN
ap-4589	66	42	algebra	algebra	NOUN
ap-4589	66	43	iso(2	iso(2	NOUN
ap-4589	66	44	)	)	PUNCT
ap-4589	66	45	of	of	ADP
ap-4589	66	46	the	the	DET
ap-4589	66	47	two	two	NUM
ap-4589	66	48	-	-	PUNCT
ap-4589	66	49	dimensional	dimensional	ADJ
ap-4589	66	50	euclidean	euclidean	ADJ
ap-4589	66	51	group	group	NOUN
ap-4589	66	52	.	.	PUNCT
ap-4589	67	1	in	in	ADP
ap-4589	67	2	this	this	DET
ap-4589	67	3	case	case	NOUN
ap-4589	67	4	the	the	DET
ap-4589	67	5	invariant	invariant	ADJ
ap-4589	67	6	operator	operator	NOUN
ap-4589	67	7	h	h	NOUN
ap-4589	67	8	is	be	AUX
ap-4589	67	9	nothing	nothing	PRON
ap-4589	67	10	but	but	SCONJ
ap-4589	67	11	the	the	DET
ap-4589	67	12	casimir	casimir	NOUN
ap-4589	67	13	operator	operator	NOUN
ap-4589	67	14	of	of	ADP
ap-4589	67	15	the	the	DET
ap-4589	67	16	lie	lie	NOUN
ap-4589	67	17	algebra	algebra	NOUN
ap-4589	67	18	iso(2	iso(2	NOUN
ap-4589	67	19	)	)	PUNCT
ap-4589	67	20	.	.	PUNCT
ap-4589	68	1	now	now	ADV
ap-4589	68	2	,	,	PUNCT
ap-4589	68	3	let	let	VERB
ap-4589	68	4	|e	|e	PROPN
ap-4589	68	5	,	,	PUNCT
ap-4589	68	6	j	j	PROPN
ap-4589	68	7	〉	〉	PROPN
ap-4589	68	8	be	be	AUX
ap-4589	68	9	a	a	DET
ap-4589	68	10	simultaneous	simultaneous	ADJ
ap-4589	68	11	eigenstate	eigenstate	NOUN
ap-4589	68	12	of	of	ADP
ap-4589	68	13	h	h	PROPN
ap-4589	68	14	and	and	CCONJ
ap-4589	68	15	j3	j3	PROPN
ap-4589	68	16	that	that	PRON
ap-4589	68	17	satisfies	satisfy	VERB
ap-4589	68	18	the	the	DET
ap-4589	68	19	eigenvalue	eigenvalue	PROPN
ap-4589	68	20	equations	equation	NOUN
ap-4589	68	21	h|e	h|e	PROPN
ap-4589	68	22	,	,	PUNCT
ap-4589	68	23	j	j	PROPN
ap-4589	68	24	〉	〉	PROPN
ap-4589	68	25	=	=	SYM
ap-4589	68	26	e|e	e|e	PROPN
ap-4589	68	27	,	,	PUNCT
ap-4589	68	28	j	j	PROPN
ap-4589	68	29	〉	〉	PROPN
ap-4589	68	30	,	,	PUNCT
ap-4589	68	31	j3|e	j3|e	PROPN
ap-4589	68	32	,	,	PUNCT
ap-4589	68	33	j	j	PROPN
ap-4589	68	34	〉	〉	PROPN
ap-4589	68	35	=	=	SYM
ap-4589	68	36	j|e	j|e	PROPN
ap-4589	68	37	,	,	PUNCT
ap-4589	68	38	j	j	PROPN
ap-4589	68	39	〉	〉	PROPN
ap-4589	68	40	,	,	PUNCT
ap-4589	68	41	(	(	PUNCT
ap-4589	68	42	2.6	2.6	NUM
ap-4589	68	43	)	)	PUNCT
ap-4589	68	44	and	and	CCONJ
ap-4589	68	45	the	the	DET
ap-4589	68	46	normalization	normalization	NOUN
ap-4589	68	47	condition	condition	NOUN
ap-4589	68	48	‖|e	‖|e	PROPN
ap-4589	68	49	,	,	PUNCT
ap-4589	68	50	j〉‖	j〉‖	NOUN
ap-4589	68	51	=	=	SYM
ap-4589	69	1	1	1	X
ap-4589	69	2	.	.	X
ap-4589	69	3	we	we	PRON
ap-4589	69	4	wish	wish	VERB
ap-4589	69	5	to	to	PART
ap-4589	69	6	find	find	VERB
ap-4589	69	7	the	the	DET
ap-4589	69	8	possible	possible	ADJ
ap-4589	69	9	values	value	NOUN
ap-4589	69	10	of	of	ADP
ap-4589	69	11	e	e	PROPN
ap-4589	69	12	and	and	CCONJ
ap-4589	69	13	j.	j.	PROPN
ap-4589	69	14	to	to	ADP
ap-4589	69	15	this	this	DET
ap-4589	69	16	end	end	NOUN
ap-4589	69	17	,	,	PUNCT
ap-4589	69	18	let	let	VERB
ap-4589	69	19	us	we	PRON
ap-4589	69	20	next	next	ADV
ap-4589	69	21	consider	consider	VERB
ap-4589	69	22	the	the	DET
ap-4589	69	23	states	state	NOUN
ap-4589	69	24	j±|e	j±|e	VERB
ap-4589	69	25	,	,	PUNCT
ap-4589	69	26	j	j	PROPN
ap-4589	69	27	〉	〉	PROPN
ap-4589	69	28	.	.	PUNCT
ap-4589	70	1	as	as	ADP
ap-4589	70	2	usual	usual	ADJ
ap-4589	70	3	,	,	PUNCT
ap-4589	70	4	the	the	DET
ap-4589	70	5	commutation	commutation	NOUN
ap-4589	70	6	relations	relation	NOUN
ap-4589	70	7	(	(	PUNCT
ap-4589	70	8	2.4	2.4	NUM
ap-4589	70	9	)	)	PUNCT
ap-4589	70	10	lead	lead	NOUN
ap-4589	70	11	j3j±|e	j3j±|e	NOUN
ap-4589	70	12	,	,	PUNCT
ap-4589	70	13	j	j	PROPN
ap-4589	70	14	〉	〉	PROPN
ap-4589	70	15	=	=	SYM
ap-4589	70	16	(	(	PUNCT
ap-4589	70	17	j	j	PROPN
ap-4589	70	18	±	±	NUM
ap-4589	70	19	1)j±|e	1)j±|e	PROPN
ap-4589	70	20	,	,	PUNCT
ap-4589	70	21	j	j	PROPN
ap-4589	70	22	〉	〉	PROPN
ap-4589	70	23	,	,	PUNCT
ap-4589	70	24	which	which	PRON
ap-4589	70	25	implies	imply	VERB
ap-4589	70	26	j±	j±	PROPN
ap-4589	70	27	raise	raise	VERB
ap-4589	70	28	and	and	CCONJ
ap-4589	70	29	lower	lower	VERB
ap-4589	70	30	the	the	DET
ap-4589	70	31	eigenvalue	eigenvalue	PROPN
ap-4589	70	32	j	j	PROPN
ap-4589	70	33	by	by	ADP
ap-4589	70	34	±1	±1	ADJ
ap-4589	70	35	:	:	PUNCT
ap-4589	70	36	j±|e	j±|e	PROPN
ap-4589	70	37	,	,	PUNCT
ap-4589	70	38	j	j	PROPN
ap-4589	70	39	〉	〉	PROPN
ap-4589	70	40	∝	∝	PROPN
ap-4589	70	41	|e	|e	PROPN
ap-4589	70	42	,	,	PUNCT
ap-4589	70	43	j	j	PROPN
ap-4589	70	44	±	±	PROPN
ap-4589	70	45	1	1	NUM
ap-4589	70	46	〉	〉	NUM
ap-4589	70	47	.	.	PUNCT
ap-4589	71	1	(	(	PUNCT
ap-4589	71	2	2.7	2.7	NUM
ap-4589	71	3	)	)	PUNCT
ap-4589	71	4	proportional	proportional	ADJ
ap-4589	71	5	coefficients	coefficient	NOUN
ap-4589	71	6	are	be	AUX
ap-4589	71	7	determined	determine	VERB
ap-4589	71	8	by	by	ADP
ap-4589	71	9	calculating	calculate	VERB
ap-4589	71	10	the	the	DET
ap-4589	71	11	norms	norm	NOUN
ap-4589	71	12	‖j±|e	‖j±|e	PROPN
ap-4589	71	13	,	,	PUNCT
ap-4589	71	14	j〉‖.	j〉‖.	ADJ
ap-4589	71	15	by	by	ADP
ap-4589	71	16	using	use	VERB
ap-4589	71	17	the	the	DET
ap-4589	71	18	relations	relation	NOUN
ap-4589	71	19	‖j±|e	‖j±|e	PROPN
ap-4589	71	20	,	,	PUNCT
ap-4589	71	21	j〉‖2	j〉‖2	PROPN
ap-4589	71	22	=	=	SYM
ap-4589	71	23	〈	〈	PROPN
ap-4589	71	24	e	e	PROPN
ap-4589	71	25	,	,	PUNCT
ap-4589	71	26	j|j∓j±|e	j|j∓j±|e	PROPN
ap-4589	71	27	,	,	PUNCT
ap-4589	71	28	j	j	PROPN
ap-4589	71	29	〉	〉	PROPN
ap-4589	71	30	,	,	PUNCT
ap-4589	71	31	j−j+	j−j+	PROPN
ap-4589	72	1	=	=	SYM
ap-4589	72	2	h	h	PROPN
ap-4589	73	1	+	+	CCONJ
ap-4589	73	2	g2	g2	PROPN
ap-4589	73	3	/	/	SYM
ap-4589	73	4	j2	j2	PROPN
ap-4589	73	5	3	3	NUM
ap-4589	73	6	,	,	PUNCT
ap-4589	73	7	and	and	CCONJ
ap-4589	73	8	j+j−	j+j−	PROPN
ap-4589	73	9	=	=	PROPN
ap-4589	73	10	h	h	PROPN
ap-4589	74	1	+	+	NUM
ap-4589	74	2	g2/(j3	g2/(j3	NOUN
ap-4589	74	3	−	−	PROPN
ap-4589	74	4	1)2	1)2	NUM
ap-4589	74	5	,	,	PUNCT
ap-4589	74	6	we	we	PRON
ap-4589	74	7	get	get	VERB
ap-4589	74	8	‖j+|e	‖j+|e	ADP
ap-4589	74	9	,	,	PUNCT
ap-4589	74	10	j〉‖2	j〉‖2	NOUN
ap-4589	74	11	=	=	SYM
ap-4589	74	12	e	e	PROPN
ap-4589	74	13	+	+	PROPN
ap-4589	74	14	g2	g2	PROPN
ap-4589	74	15	j2	j2	PROPN
ap-4589	74	16	≥	≥	PROPN
ap-4589	74	17	0	0	NUM
ap-4589	74	18	,	,	PUNCT
ap-4589	74	19	‖j−|e	‖j−|e	NUM
ap-4589	74	20	,	,	PUNCT
ap-4589	74	21	j〉‖2	j〉‖2	NOUN
ap-4589	74	22	=	=	SYM
ap-4589	74	23	e	e	PROPN
ap-4589	74	24	+	+	CCONJ
ap-4589	74	25	g2	g2	PROPN
ap-4589	74	26	(	(	PUNCT
ap-4589	74	27	j	j	PROPN
ap-4589	75	1	−	−	PROPN
ap-4589	75	2	1)2	1)2	NUM
ap-4589	75	3	≥	≥	NOUN
ap-4589	75	4	0	0	NUM
ap-4589	75	5	.	.	PUNCT
ap-4589	76	1	(	(	PUNCT
ap-4589	76	2	2.8	2.8	NUM
ap-4589	76	3	)	)	PUNCT
ap-4589	76	4	these	these	DET
ap-4589	76	5	equations	equation	NOUN
ap-4589	76	6	not	not	PART
ap-4589	76	7	only	only	ADV
ap-4589	76	8	fix	fix	VERB
ap-4589	76	9	the	the	DET
ap-4589	76	10	proportional	proportional	ADJ
ap-4589	76	11	coefficients	coefficient	NOUN
ap-4589	76	12	in	in	ADP
ap-4589	76	13	(	(	PUNCT
ap-4589	76	14	2.7	2.7	NUM
ap-4589	76	15	)	)	PUNCT
ap-4589	76	16	but	but	CCONJ
ap-4589	76	17	also	also	ADV
ap-4589	76	18	provide	provide	VERB
ap-4589	76	19	nontrivial	nontrivial	ADJ
ap-4589	76	20	constraints	constraint	NOUN
ap-4589	76	21	on	on	ADP
ap-4589	76	22	e	e	PROPN
ap-4589	76	23	and	and	CCONJ
ap-4589	76	24	j.	j.	PROPN
ap-4589	76	25	in	in	ADP
ap-4589	76	26	fact	fact	NOUN
ap-4589	76	27	,	,	PUNCT
ap-4589	76	28	together	together	ADV
ap-4589	76	29	with	with	ADP
ap-4589	76	30	the	the	DET
ap-4589	76	31	ladder	ladder	NOUN
ap-4589	76	32	equations	equation	NOUN
ap-4589	76	33	(	(	PUNCT
ap-4589	76	34	2.7	2.7	NUM
ap-4589	76	35	)	)	PUNCT
ap-4589	76	36	,	,	PUNCT
ap-4589	76	37	the	the	DET
ap-4589	76	38	conditions	condition	NOUN
ap-4589	76	39	(	(	PUNCT
ap-4589	76	40	2.8	2.8	NUM
ap-4589	76	41	)	)	PUNCT
ap-4589	76	42	completely	completely	ADV
ap-4589	76	43	fix	fix	VERB
ap-4589	76	44	the	the	DET
ap-4589	76	45	possible	possible	ADJ
ap-4589	76	46	values	value	NOUN
ap-4589	76	47	of	of	ADP
ap-4589	76	48	e	e	PROPN
ap-4589	76	49	and	and	CCONJ
ap-4589	76	50	j.	j.	PROPN
ap-4589	76	51	to	to	PART
ap-4589	76	52	see	see	VERB
ap-4589	76	53	this	this	PRON
ap-4589	76	54	,	,	PUNCT
ap-4589	76	55	let	let	VERB
ap-4589	76	56	us	we	PRON
ap-4589	76	57	consider	consider	VERB
ap-4589	76	58	a	a	DET
ap-4589	76	59	negative	negative	ADJ
ap-4589	76	60	-	-	PUNCT
ap-4589	76	61	energy	energy	NOUN
ap-4589	76	62	state	state	NOUN
ap-4589	76	63	|e	|e	PROPN
ap-4589	76	64	,	,	PUNCT
ap-4589	76	65	j	j	PROPN
ap-4589	76	66	〉	〉	NOUN
ap-4589	76	67	that	that	PRON
ap-4589	76	68	corresponds	correspond	VERB
ap-4589	76	69	to	to	ADP
ap-4589	76	70	an	an	DET
ap-4589	76	71	arbitrary	arbitrary	ADJ
ap-4589	76	72	point	point	NOUN
ap-4589	76	73	in	in	ADP
ap-4589	76	74	the	the	DET
ap-4589	76	75	lower	low	ADJ
ap-4589	76	76	half	half	NOUN
ap-4589	76	77	of	of	ADP
ap-4589	76	78	the	the	DET
ap-4589	76	79	(	(	PUNCT
ap-4589	76	80	e	e	NOUN
ap-4589	76	81	,	,	PUNCT
ap-4589	76	82	j)-plane	j)-plane	NOUN
ap-4589	76	83	.	.	PUNCT
ap-4589	77	1	by	by	ADP
ap-4589	77	2	applying	apply	VERB
ap-4589	77	3	the	the	DET
ap-4589	77	4	ladder	ladder	NOUN
ap-4589	77	5	operators	operator	NOUN
ap-4589	77	6	j±	j±	PROPN
ap-4589	77	7	to	to	ADP
ap-4589	77	8	the	the	DET
ap-4589	77	9	state	state	NOUN
ap-4589	77	10	|e	|e	PROPN
ap-4589	77	11	,	,	PUNCT
ap-4589	77	12	j	j	PROPN
ap-4589	77	13	〉	〉	PROPN
ap-4589	77	14	one	one	NOUN
ap-4589	77	15	can	can	AUX
ap-4589	77	16	easily	easily	ADV
ap-4589	77	17	see	see	VERB
ap-4589	77	18	that	that	SCONJ
ap-4589	77	19	such	such	DET
ap-4589	77	20	an	an	DET
ap-4589	77	21	arbitrary	arbitrary	ADJ
ap-4589	77	22	point	point	NOUN
ap-4589	77	23	eventually	eventually	ADV
ap-4589	77	24	falls	fall	VERB
ap-4589	77	25	into	into	ADP
ap-4589	77	26	the	the	DET
ap-4589	77	27	region	region	NOUN
ap-4589	77	28	in	in	ADP
ap-4589	77	29	which	which	PRON
ap-4589	77	30	the	the	DET
ap-4589	77	31	squared	squared	ADJ
ap-4589	77	32	norms	norm	NOUN
ap-4589	77	33	become	become	VERB
ap-4589	77	34	negative	negative	ADJ
ap-4589	77	35	.	.	PUNCT
ap-4589	78	1	see	see	VERB
ap-4589	78	2	the	the	DET
ap-4589	78	3	figure	figure	NOUN
ap-4589	78	4	below	below	ADV
ap-4589	78	5	:	:	PUNCT
ap-4589	78	6	j	j	PROPN
ap-4589	78	7	e	e	NOUN
ap-4589	78	8	1	1	NUM
ap-4589	78	9	2	2	NUM
ap-4589	78	10	|e	|e	PROPN
ap-4589	78	11	,	,	PUNCT
ap-4589	78	12	j〉j+	j〉j+	NUM
ap-4589	78	13	j−	j−	PROPN
ap-4589	78	14	e	e	NOUN
ap-4589	78	15	=	=	PUNCT
ap-4589	79	1	−	−	PROPN
ap-4589	79	2	g2	g2	PROPN
ap-4589	79	3	j2	j2	PROPN
ap-4589	79	4	e	e	PROPN
ap-4589	79	5	=	=	PROPN
ap-4589	79	6	−	−	PROPN
ap-4589	79	7	g2	g2	PROPN
ap-4589	79	8	(	(	PUNCT
ap-4589	79	9	j	j	PROPN
ap-4589	79	10	−	−	PROPN
ap-4589	79	11	1)2	1)2	NUM
ap-4589	79	12	‖j±|e	‖j±|e	PROPN
ap-4589	79	13	,	,	PUNCT
ap-4589	79	14	j〉‖2	j〉‖2	NOUN
ap-4589	79	15	<	<	X
ap-4589	79	16	0	0	PUNCT
ap-4589	80	1	the	the	DET
ap-4589	80	2	only	only	ADJ
ap-4589	80	3	way	way	NOUN
ap-4589	80	4	to	to	PART
ap-4589	80	5	avoid	avoid	VERB
ap-4589	80	6	this	this	PRON
ap-4589	80	7	is	be	AUX
ap-4589	80	8	to	to	PART
ap-4589	80	9	terminate	terminate	VERB
ap-4589	80	10	the	the	DET
ap-4589	80	11	sequence	sequence	NOUN
ap-4589	80	12	{	{	PUNCT
ap-4589	80	13	·	·	PUNCT
ap-4589	80	14	·	·	PUNCT
ap-4589	80	15	·	·	PUNCT
ap-4589	80	16	,	,	PUNCT
ap-4589	80	17	|e	|e	PROPN
ap-4589	80	18	,	,	PUNCT
ap-4589	80	19	j−1	j−1	PROPN
ap-4589	80	20	〉	〉	PROPN
ap-4589	80	21	,	,	PUNCT
ap-4589	80	22	|e	|e	PROPN
ap-4589	80	23	,	,	PUNCT
ap-4589	80	24	j	j	PROPN
ap-4589	80	25	〉	〉	PROPN
ap-4589	80	26	,	,	PUNCT
ap-4589	80	27	|e	|e	PROPN
ap-4589	80	28	,	,	PUNCT
ap-4589	80	29	j+1	j+1	PROPN
ap-4589	80	30	〉	〉	NUM
ap-4589	80	31	,	,	PUNCT
ap-4589	80	32	·	·	PUNCT
ap-4589	80	33	·	·	PUNCT
ap-4589	80	34	·	·	PUNCT
ap-4589	80	35	}	}	PUNCT
ap-4589	80	36	from	from	ADP
ap-4589	80	37	both	both	CCONJ
ap-4589	80	38	above	above	ADP
ap-4589	80	39	and	and	CCONJ
ap-4589	80	40	below	below	ADV
ap-4589	80	41	.	.	PUNCT
ap-4589	81	1	this	this	PRON
ap-4589	81	2	is	be	AUX
ap-4589	81	3	possible	possible	ADJ
ap-4589	81	4	if	if	SCONJ
ap-4589	81	5	and	and	CCONJ
ap-4589	81	6	only	only	ADV
ap-4589	81	7	if	if	SCONJ
ap-4589	81	8	there	there	PRON
ap-4589	81	9	exist	exist	VERB
ap-4589	81	10	both	both	CCONJ
ap-4589	81	11	the	the	DET
ap-4589	81	12	highest	high	ADJ
ap-4589	81	13	and	and	CCONJ
ap-4589	81	14	lowest	low	ADJ
ap-4589	81	15	weight	weight	NOUN
ap-4589	81	16	states	state	NOUN
ap-4589	81	17	|e	|e	PROPN
ap-4589	81	18	,	,	PUNCT
ap-4589	81	19	jmax	jmax	PROPN
ap-4589	81	20	〉	〉	PROPN
ap-4589	81	21	and	and	CCONJ
ap-4589	81	22	|e	|e	PROPN
ap-4589	81	23	,	,	PUNCT
ap-4589	81	24	jmin	jmin	ADJ
ap-4589	81	25	〉	〉	PROPN
ap-4589	81	26	in	in	ADP
ap-4589	81	27	the	the	DET
ap-4589	81	28	sequence	sequence	NOUN
ap-4589	81	29	such	such	ADJ
ap-4589	81	30	that	that	DET
ap-4589	81	31	j+|e	j+|e	PROPN
ap-4589	81	32	,	,	PUNCT
ap-4589	81	33	jmax	jmax	PROPN
ap-4589	81	34	〉	〉	NOUN
ap-4589	81	35	=	=	SYM
ap-4589	81	36	0	0	PUNCT
ap-4589	81	37	=	=	SYM
ap-4589	81	38	j−|e	j−|e	PROPN
ap-4589	81	39	,	,	PUNCT
ap-4589	81	40	jmin	jmin	PROPN
ap-4589	81	41	〉	〉	PROPN
ap-4589	81	42	,	,	PUNCT
ap-4589	81	43	−g2	−g2	PROPN
ap-4589	81	44	/	/	SYM
ap-4589	81	45	j2	j2	PROPN
ap-4589	81	46	max	max	PROPN
ap-4589	82	1	=	=	PROPN
ap-4589	83	1	−g2/(jmin	−g2/(jmin	PROPN
ap-4589	84	1	−	−	PROPN
ap-4589	84	2	1)2	1)2	NUM
ap-4589	84	3	,	,	PUNCT
ap-4589	84	4	jmax	jmax	PROPN
ap-4589	84	5	−	−	PROPN
ap-4589	84	6	jmin	jmin	PROPN
ap-4589	84	7	∈	∈	PROPN
ap-4589	84	8	z≥0	z≥0	PROPN
ap-4589	84	9	,	,	PUNCT
ap-4589	84	10	and	and	CCONJ
ap-4589	84	11	jmax	jmax	PROPN
ap-4589	84	12	≥	≥	NUM
ap-4589	84	13	1/2	1/2	NUM
ap-4589	84	14	and	and	CCONJ
ap-4589	84	15	jmin	jmin	ADJ
ap-4589	84	16	≤	≤	NUM
ap-4589	84	17	1/2	1/2	NUM
ap-4589	84	18	.	.	PUNCT
ap-4589	85	1	it	it	PRON
ap-4589	85	2	is	be	AUX
ap-4589	85	3	not	not	PART
ap-4589	85	4	difficult	difficult	ADJ
ap-4589	85	5	to	to	PART
ap-4589	85	6	see	see	VERB
ap-4589	85	7	that	that	SCONJ
ap-4589	85	8	these	these	DET
ap-4589	85	9	conditions	condition	NOUN
ap-4589	85	10	are	be	AUX
ap-4589	85	11	fulfilled	fulfil	VERB
ap-4589	85	12	if	if	SCONJ
ap-4589	85	13	and	and	CCONJ
ap-4589	85	14	only	only	ADV
ap-4589	85	15	if	if	SCONJ
ap-4589	85	16	the	the	DET
ap-4589	85	17	eigenvalue	eigenvalue	NOUN
ap-4589	85	18	of	of	ADP
ap-4589	85	19	the	the	DET
ap-4589	85	20	invariant	invariant	ADJ
ap-4589	85	21	operator	operator	NOUN
ap-4589	85	22	takes	take	VERB
ap-4589	85	23	the	the	DET
ap-4589	85	24	value	value	NOUN
ap-4589	85	25	e	e	NOUN
ap-4589	85	26	=	=	SYM
ap-4589	85	27	−g2	−g2	ADJ
ap-4589	85	28	/	/	SYM
ap-4589	85	29	ν2	ν2	NOUN
ap-4589	85	30	,	,	PUNCT
ap-4589	85	31	ν	ν	X
ap-4589	85	32	∈	∈	PROPN
ap-4589	85	33	{	{	PUNCT
ap-4589	85	34	1	1	NUM
ap-4589	85	35	2	2	NUM
ap-4589	85	36	,	,	PUNCT
ap-4589	85	37	1	1	NUM
ap-4589	85	38	,	,	PUNCT
ap-4589	85	39	3	3	NUM
ap-4589	85	40	2	2	NUM
ap-4589	85	41	,	,	PUNCT
ap-4589	85	42	2	2	NUM
ap-4589	85	43	,	,	PUNCT
ap-4589	85	44	·	·	PUNCT
ap-4589	85	45	·	·	PUNCT
ap-4589	85	46	·	·	PUNCT
ap-4589	85	47	}	}	PUNCT
ap-4589	85	48	.	.	PUNCT
ap-4589	86	1	with	with	ADP
ap-4589	86	2	this	this	DET
ap-4589	86	3	ν	ν	ADP
ap-4589	86	4	the	the	DET
ap-4589	86	5	eigenvalues	eigenvalue	NOUN
ap-4589	86	6	of	of	ADP
ap-4589	86	7	j3	j3	PROPN
ap-4589	86	8	take	take	VERB
ap-4589	86	9	the	the	DET
ap-4589	86	10	values	value	NOUN
ap-4589	86	11	{	{	PUNCT
ap-4589	86	12	jmax	jmax	NOUN
ap-4589	86	13	=	=	SYM
ap-4589	86	14	ν	ν	NOUN
ap-4589	86	15	,	,	PUNCT
ap-4589	86	16	ν	ν	X
ap-4589	86	17	−	−	PROPN
ap-4589	86	18	1	1	NUM
ap-4589	86	19	,	,	PUNCT
ap-4589	86	20	·	·	PUNCT
ap-4589	86	21	·	·	PUNCT
ap-4589	86	22	·	·	PUNCT
ap-4589	86	23	,	,	PUNCT
ap-4589	86	24	2−	2−	NUM
ap-4589	86	25	ν	ν	NOUN
ap-4589	86	26	,	,	PUNCT
ap-4589	86	27	jmin	jmin	NOUN
ap-4589	86	28	=	=	SYM
ap-4589	86	29	1−	1−	NUM
ap-4589	86	30	ν	ν	NOUN
ap-4589	86	31	}	}	PUNCT
ap-4589	86	32	.	.	PUNCT
ap-4589	87	1	note	note	NOUN
ap-4589	87	2	,	,	PUNCT
ap-4589	87	3	however	however	ADV
ap-4589	87	4	,	,	PUNCT
ap-4589	87	5	that	that	SCONJ
ap-4589	87	6	if	if	SCONJ
ap-4589	87	7	ν	ν	NOUN
ap-4589	87	8	is	be	AUX
ap-4589	87	9	an	an	DET
ap-4589	87	10	integer	integer	NOUN
ap-4589	87	11	,	,	PUNCT
ap-4589	87	12	the	the	DET
ap-4589	87	13	spectrum	spectrum	NOUN
ap-4589	87	14	of	of	ADP
ap-4589	87	15	j3	j3	PROPN
ap-4589	87	16	contains	contain	VERB
ap-4589	87	17	j	j	PROPN
ap-4589	87	18	=	=	SYM
ap-4589	87	19	0	0	NUM
ap-4589	87	20	which	which	PRON
ap-4589	87	21	makes	make	VERB
ap-4589	87	22	the	the	DET
ap-4589	87	23	operator	operator	NOUN
ap-4589	87	24	1	1	NUM
ap-4589	87	25	/	/	SYM
ap-4589	87	26	j3	j3	PROPN
ap-4589	87	27	ill	ill	ADV
ap-4589	87	28	-	-	PUNCT
ap-4589	87	29	defined	define	VERB
ap-4589	87	30	.	.	PUNCT
ap-4589	88	1	thus	thus	ADV
ap-4589	88	2	we	we	PRON
ap-4589	88	3	should	should	AUX
ap-4589	88	4	disregard	disregard	VERB
ap-4589	88	5	this	this	DET
ap-4589	88	6	case	case	NOUN
ap-4589	88	7	.	.	PUNCT
ap-4589	89	1	to	to	PART
ap-4589	89	2	summarize	summarize	VERB
ap-4589	89	3	,	,	PUNCT
ap-4589	89	4	4the	4the	NUM
ap-4589	89	5	commutation	commutation	NOUN
ap-4589	89	6	relation	relation	NOUN
ap-4589	89	7	[	[	X
ap-4589	89	8	h	h	X
ap-4589	89	9	,	,	PUNCT
ap-4589	89	10	j3	j3	PROPN
ap-4589	89	11	]	]	X
ap-4589	89	12	=	=	SYM
ap-4589	89	13	0	0	NUM
ap-4589	89	14	is	be	AUX
ap-4589	89	15	trivial	trivial	ADJ
ap-4589	89	16	.	.	PUNCT
ap-4589	90	1	in	in	ADP
ap-4589	90	2	order	order	NOUN
ap-4589	90	3	to	to	PART
ap-4589	90	4	prove	prove	VERB
ap-4589	90	5	[	[	X
ap-4589	90	6	h	h	NOUN
ap-4589	90	7	,	,	PUNCT
ap-4589	90	8	j±	j±	PROPN
ap-4589	90	9	]	]	X
ap-4589	90	10	=	=	SYM
ap-4589	90	11	0	0	NUM
ap-4589	90	12	,	,	PUNCT
ap-4589	90	13	one	one	PRON
ap-4589	90	14	should	should	AUX
ap-4589	90	15	first	first	ADV
ap-4589	90	16	note	note	VERB
ap-4589	90	17	that	that	SCONJ
ap-4589	90	18	hj+−j+h	hj+−j+h	PROPN
ap-4589	90	19	=	=	SYM
ap-4589	90	20	g2(j+	g2(j+	PROPN
ap-4589	90	21	1	1	NUM
ap-4589	90	22	j2	j2	NOUN
ap-4589	90	23	3	3	NUM
ap-4589	90	24	−	−	NOUN
ap-4589	90	25	1	1	NUM
ap-4589	90	26	(	(	PUNCT
ap-4589	90	27	j3−1)2	j3−1)2	PROPN
ap-4589	90	28	j+	j+	NUM
ap-4589	90	29	)	)	PUNCT
ap-4589	90	30	and	and	CCONJ
ap-4589	90	31	hj−	hj−	PROPN
ap-4589	90	32	−	−	PROPN
ap-4589	90	33	j−h	j−h	PROPN
ap-4589	90	34	=	=	SYM
ap-4589	90	35	g2(j−	g2(j−	X
ap-4589	90	36	1	1	X
ap-4589	90	37	(	(	PUNCT
ap-4589	90	38	j3−1)2	j3−1)2	X
ap-4589	90	39	−	−	PROPN
ap-4589	90	40	1	1	NUM
ap-4589	90	41	j2	j2	PROPN
ap-4589	90	42	3	3	NUM
ap-4589	90	43	j−	j−	PROPN
ap-4589	90	44	)	)	PUNCT
ap-4589	90	45	,	,	PUNCT
ap-4589	90	46	which	which	PRON
ap-4589	90	47	follow	follow	VERB
ap-4589	90	48	from	from	ADP
ap-4589	90	49	(	(	PUNCT
ap-4589	90	50	2.5	2.5	NUM
ap-4589	90	51	)	)	PUNCT
ap-4589	90	52	.	.	PUNCT
ap-4589	91	1	then	then	ADV
ap-4589	91	2	by	by	ADP
ap-4589	91	3	using	use	VERB
ap-4589	91	4	(	(	PUNCT
ap-4589	91	5	2.3	2.3	NUM
ap-4589	91	6	)	)	PUNCT
ap-4589	91	7	and	and	CCONJ
ap-4589	91	8	e−iθ	e−iθ	X
ap-4589	91	9	1	1	NUM
ap-4589	91	10	(	(	PUNCT
ap-4589	91	11	j3−1)2	j3−1)2	PROPN
ap-4589	91	12	eiθ	eiθ	PROPN
ap-4589	91	13	=	=	SYM
ap-4589	91	14	1	1	NUM
ap-4589	91	15	j2	j2	NOUN
ap-4589	91	16	3	3	NUM
ap-4589	91	17	,	,	PUNCT
ap-4589	91	18	one	one	NUM
ap-4589	91	19	arrives	arrive	VERB
ap-4589	91	20	at	at	ADP
ap-4589	91	21	[	[	X
ap-4589	91	22	h	h	NOUN
ap-4589	91	23	,	,	PUNCT
ap-4589	91	24	j±	j±	PROPN
ap-4589	91	25	]	]	X
ap-4589	91	26	=	=	SYM
ap-4589	91	27	0	0	X
ap-4589	91	28	.	.	PUNCT
ap-4589	92	1	448	448	NUM
ap-4589	92	2	vol	vol	NOUN
ap-4589	92	3	.	.	PUNCT
ap-4589	92	4	57	57	NUM
ap-4589	92	5	no	no	NOUN
ap-4589	92	6	.	.	PUNCT
ap-4589	93	1	6/2017	6/2017	X
ap-4589	93	2	algebraic	algebraic	ADJ
ap-4589	93	3	description	description	NOUN
ap-4589	93	4	of	of	ADP
ap-4589	93	5	shape	shape	NOUN
ap-4589	93	6	invariance	invariance	NOUN
ap-4589	93	7	revisited	revisit	VERB
ap-4589	93	8	·	·	PUNCT
ap-4589	93	9	·	·	PUNCT
ap-4589	93	10	·	·	PUNCT
ap-4589	93	11	·	·	PUNCT
ap-4589	93	12	·	·	PUNCT
ap-4589	93	13	·	·	PUNCT
ap-4589	93	14	·	·	PUNCT
ap-4589	93	15	·	·	PUNCT
ap-4589	93	16	·	·	PUNCT
ap-4589	93	17	·	·	PUNCT
ap-4589	93	18	·	·	PUNCT
ap-4589	93	19	·	·	PUNCT
ap-4589	93	20	·	·	PUNCT
ap-4589	93	21	·	·	PUNCT
ap-4589	93	22	·	·	PUNCT
ap-4589	94	1	j	j	PROPN
ap-4589	94	2	e	e	NOUN
ap-4589	94	3	−	−	PROPN
ap-4589	94	4	3	3	NUM
ap-4589	94	5	2	2	NUM
ap-4589	94	6	−	−	NUM
ap-4589	94	7	1	1	NUM
ap-4589	94	8	2	2	NUM
ap-4589	94	9	1	1	NUM
ap-4589	94	10	2	2	NUM
ap-4589	94	11	3	3	NUM
ap-4589	94	12	2	2	NUM
ap-4589	94	13	5	5	NUM
ap-4589	94	14	2	2	NUM
ap-4589	94	15	(	(	PUNCT
ap-4589	94	16	a	a	NOUN
ap-4589	94	17	)	)	PUNCT
ap-4589	94	18	kepler	kepler	NOUN
ap-4589	94	19	·	·	PUNCT
ap-4589	94	20	·	·	PUNCT
ap-4589	94	21	·	·	PUNCT
ap-4589	94	22	·	·	PUNCT
ap-4589	94	23	·	·	PUNCT
ap-4589	94	24	·	·	PUNCT
ap-4589	94	25	·	·	PUNCT
ap-4589	94	26	·	·	PUNCT
ap-4589	94	27	·	·	PUNCT
ap-4589	94	28	·	·	PUNCT
ap-4589	94	29	·	·	PUNCT
ap-4589	94	30	·	·	PUNCT
ap-4589	94	31	·	·	PUNCT
ap-4589	94	32	·	·	PUNCT
ap-4589	94	33	·	·	PUNCT
ap-4589	94	34	j	j	PROPN
ap-4589	95	1	e	e	NOUN
ap-4589	95	2	−	−	PROPN
ap-4589	95	3	3	3	NUM
ap-4589	95	4	2	2	NUM
ap-4589	95	5	−	−	NUM
ap-4589	95	6	1	1	NUM
ap-4589	95	7	2	2	NUM
ap-4589	95	8	1	1	NUM
ap-4589	95	9	2	2	NUM
ap-4589	95	10	3	3	NUM
ap-4589	95	11	2	2	NUM
ap-4589	95	12	5	5	NUM
ap-4589	95	13	2	2	NUM
ap-4589	95	14	(	(	PUNCT
ap-4589	95	15	b	b	NOUN
ap-4589	95	16	)	)	PUNCT
ap-4589	95	17	spherical	spherical	ADJ
ap-4589	95	18	kepler	kepler	NOUN
ap-4589	95	19	·	·	PUNCT
ap-4589	95	20	·	·	PUNCT
ap-4589	95	21	·	·	PUNCT
ap-4589	95	22	·	·	PUNCT
ap-4589	95	23	·	·	PUNCT
ap-4589	95	24	·	·	PUNCT
ap-4589	95	25	·	·	PUNCT
ap-4589	95	26	·	·	PUNCT
ap-4589	95	27	·	·	PUNCT
ap-4589	95	28	·	·	PUNCT
ap-4589	95	29	·	·	PUNCT
ap-4589	95	30	·	·	PUNCT
ap-4589	95	31	·	·	PUNCT
ap-4589	95	32	·	·	PUNCT
ap-4589	95	33	·	·	PUNCT
ap-4589	95	34	·	·	PUNCT
ap-4589	95	35	·	·	PUNCT
ap-4589	95	36	·	·	PUNCT
ap-4589	95	37	·	·	PUNCT
ap-4589	95	38	·	·	PUNCT
ap-4589	95	39	·	·	PUNCT
ap-4589	95	40	j	j	PROPN
ap-4589	95	41	e	e	ADP
ap-4589	95	42	1	1	NUM
ap-4589	95	43	2	2	NUM
ap-4589	95	44	−√g	−√g	NUM
ap-4589	95	45	1	1	NUM
ap-4589	95	46	2	2	NUM
ap-4589	95	47	−	−	NOUN
ap-4589	95	48	√	√	NOUN
ap-4589	95	49	1	1	NUM
ap-4589	95	50	4	4	NUM
ap-4589	95	51	+	+	CCONJ
ap-4589	95	52	g	g	NOUN
ap-4589	95	53	1−√g	1−√g	NUM
ap-4589	95	54	√	√	NOUN
ap-4589	95	55	g	g	NOUN
ap-4589	95	56	1	1	NUM
ap-4589	95	57	2	2	NUM
ap-4589	95	58	+	+	CCONJ
ap-4589	95	59	√	√	NUM
ap-4589	95	60	1	1	NUM
ap-4589	95	61	4	4	NUM
ap-4589	95	62	+	+	CCONJ
ap-4589	95	63	g	g	NOUN
ap-4589	95	64	1	1	NUM
ap-4589	95	65	+	+	NOUN
ap-4589	95	66	√g	√g	X
ap-4589	95	67	(	(	PUNCT
ap-4589	95	68	c	c	NOUN
ap-4589	95	69	)	)	PUNCT
ap-4589	95	70	hyperbolic	hyperbolic	ADJ
ap-4589	95	71	kepler	kepler	PROPN
ap-4589	95	72	&	&	CCONJ
ap-4589	95	73	rosen	rosen	PROPN
ap-4589	95	74	–	–	PUNCT
ap-4589	95	75	morse	morse	PROPN
ap-4589	95	76	:	:	PUNCT
ap-4589	95	77	case	case	NOUN
ap-4589	95	78	g	g	PROPN
ap-4589	95	79	>	>	PROPN
ap-4589	95	80	1/4	1/4	NUM
ap-4589	95	81	figure	figure	NOUN
ap-4589	95	82	2	2	NUM
ap-4589	95	83	.	.	PUNCT
ap-4589	95	84	representations	representation	NOUN
ap-4589	95	85	of	of	ADP
ap-4589	95	86	the	the	DET
ap-4589	95	87	potential	potential	ADJ
ap-4589	95	88	algebras	algebra	NOUN
ap-4589	95	89	.	.	PUNCT
ap-4589	96	1	gray	gray	ADJ
ap-4589	96	2	shaded	shade	VERB
ap-4589	96	3	regions	region	NOUN
ap-4589	96	4	are	be	AUX
ap-4589	96	5	the	the	DET
ap-4589	96	6	domains	domain	NOUN
ap-4589	96	7	in	in	ADP
ap-4589	96	8	which	which	PRON
ap-4589	96	9	the	the	DET
ap-4589	96	10	squared	squared	ADJ
ap-4589	96	11	norms	norm	NOUN
ap-4589	96	12	‖j±|e	‖j±|e	PROPN
ap-4589	96	13	,	,	PUNCT
ap-4589	96	14	j〉‖2	j〉‖2	PROPN
ap-4589	96	15	become	become	VERB
ap-4589	96	16	negative	negative	ADJ
ap-4589	96	17	.	.	PUNCT
ap-4589	97	1	red	red	ADJ
ap-4589	97	2	circles	circle	NOUN
ap-4589	97	3	represent	represent	VERB
ap-4589	97	4	the	the	DET
ap-4589	97	5	finite	finite	ADJ
ap-4589	97	6	-	-	ADJ
ap-4589	97	7	dimensional	dimensional	ADJ
ap-4589	97	8	representations	representation	NOUN
ap-4589	97	9	,	,	PUNCT
ap-4589	97	10	whereas	whereas	SCONJ
ap-4589	97	11	blue	blue	ADJ
ap-4589	97	12	circles	circle	NOUN
ap-4589	97	13	represent	represent	VERB
ap-4589	97	14	the	the	DET
ap-4589	97	15	infinite	infinite	ADJ
ap-4589	97	16	-	-	PUNCT
ap-4589	97	17	dimensional	dimensional	ADJ
ap-4589	97	18	representations	representation	NOUN
ap-4589	97	19	.	.	PUNCT
ap-4589	98	1	right	right	ADJ
ap-4589	98	2	and	and	CCONJ
ap-4589	98	3	left	leave	VERB
ap-4589	98	4	arrows	arrow	NOUN
ap-4589	98	5	indicate	indicate	VERB
ap-4589	98	6	the	the	DET
ap-4589	98	7	actions	action	NOUN
ap-4589	98	8	of	of	ADP
ap-4589	98	9	ladder	ladder	NOUN
ap-4589	98	10	operators	operator	NOUN
ap-4589	98	11	j+	j+	NUM
ap-4589	98	12	and	and	CCONJ
ap-4589	98	13	j−	j−	PROPN
ap-4589	98	14	,	,	PUNCT
ap-4589	98	15	respectively	respectively	ADV
ap-4589	98	16	.	.	PUNCT
ap-4589	99	1	the	the	DET
ap-4589	99	2	representation	representation	NOUN
ap-4589	99	3	of	of	ADP
ap-4589	99	4	the	the	DET
ap-4589	99	5	potential	potential	ADJ
ap-4589	99	6	algebra	algebra	NOUN
ap-4589	99	7	is	be	AUX
ap-4589	99	8	specified	specify	VERB
ap-4589	99	9	by	by	ADP
ap-4589	99	10	a	a	DET
ap-4589	99	11	half	half	ADJ
ap-4589	99	12	-	-	PUNCT
ap-4589	99	13	integer	integer	NOUN
ap-4589	99	14	ν	ν	X
ap-4589	99	15	∈	∈	PROPN
ap-4589	99	16	{	{	PUNCT
ap-4589	99	17	1	1	NUM
ap-4589	99	18	2	2	NUM
ap-4589	99	19	,	,	PUNCT
ap-4589	99	20	3	3	NUM
ap-4589	99	21	2	2	NUM
ap-4589	99	22	,	,	PUNCT
ap-4589	99	23	·	·	PUNCT
ap-4589	99	24	·	·	PUNCT
ap-4589	99	25	·	·	PUNCT
ap-4589	99	26	}	}	PUNCT
ap-4589	99	27	and	and	CCONJ
ap-4589	99	28	the	the	DET
ap-4589	99	29	representation	representation	NOUN
ap-4589	99	30	space	space	NOUN
ap-4589	99	31	is	be	AUX
ap-4589	99	32	spanned	span	VERB
ap-4589	99	33	by	by	ADP
ap-4589	99	34	the	the	DET
ap-4589	99	35	following	follow	VERB
ap-4589	99	36	2ν	2ν	NUM
ap-4589	99	37	vectors	vector	NOUN
ap-4589	99	38	:	:	PUNCT
ap-4589	99	39	{	{	PUNCT
ap-4589	99	40	|e	|e	PROPN
ap-4589	99	41	,	,	PUNCT
ap-4589	99	42	j	j	PROPN
ap-4589	99	43	〉	〉	NOUN
ap-4589	99	44	:	:	PUNCT
ap-4589	100	1	e	e	X
ap-4589	100	2	=	=	PRON
ap-4589	100	3	−g	−g	VERB
ap-4589	100	4	2	2	NUM
ap-4589	100	5	ν2	ν2	NOUN
ap-4589	100	6	and	and	CCONJ
ap-4589	100	7	j	j	PROPN
ap-4589	100	8	∈	∈	PROPN
ap-4589	100	9	{	{	PUNCT
ap-4589	100	10	ν	ν	NOUN
ap-4589	100	11	,	,	PUNCT
ap-4589	100	12	ν	ν	X
ap-4589	100	13	−	−	PROPN
ap-4589	100	14	1	1	NUM
ap-4589	100	15	,	,	PUNCT
ap-4589	100	16	·	·	PUNCT
ap-4589	100	17	·	·	PUNCT
ap-4589	100	18	·	·	PUNCT
ap-4589	100	19	,	,	PUNCT
ap-4589	100	20	1−	1−	NUM
ap-4589	100	21	ν	ν	X
ap-4589	100	22	}	}	PUNCT
ap-4589	100	23	}	}	PUNCT
ap-4589	100	24	.	.	PUNCT
ap-4589	101	1	(	(	PUNCT
ap-4589	101	2	2.9	2.9	NUM
ap-4589	101	3	)	)	PUNCT
ap-4589	101	4	these	these	DET
ap-4589	101	5	2ν	2ν	ADV
ap-4589	101	6	-	-	PUNCT
ap-4589	101	7	dimensional	dimensional	ADJ
ap-4589	101	8	representations	representation	NOUN
ap-4589	101	9	are	be	AUX
ap-4589	101	10	schematically	schematically	ADV
ap-4589	101	11	depicted	depict	VERB
ap-4589	101	12	in	in	ADP
ap-4589	101	13	figure	figure	NOUN
ap-4589	101	14	2(a	2(a	NUM
ap-4589	101	15	)	)	PUNCT
ap-4589	101	16	.	.	PUNCT
ap-4589	102	1	now	now	ADV
ap-4589	102	2	it	it	PRON
ap-4589	102	3	is	be	AUX
ap-4589	102	4	straightforward	straightforward	ADJ
ap-4589	102	5	to	to	PART
ap-4589	102	6	solve	solve	VERB
ap-4589	102	7	the	the	DET
ap-4589	102	8	original	original	ADJ
ap-4589	102	9	spectral	spectral	ADJ
ap-4589	102	10	problem	problem	NOUN
ap-4589	102	11	of	of	ADP
ap-4589	102	12	the	the	DET
ap-4589	102	13	kepler	kepler	PROPN
ap-4589	102	14	hamiltonian	hamiltonian	PROPN
ap-4589	102	15	hkepler	hkepler	NOUN
ap-4589	102	16	.	.	PUNCT
ap-4589	103	1	to	to	ADP
ap-4589	103	2	this	this	DET
ap-4589	103	3	end	end	NOUN
ap-4589	103	4	,	,	PUNCT
ap-4589	103	5	let	let	VERB
ap-4589	103	6	j	j	PROPN
ap-4589	103	7	∈	∈	PROPN
ap-4589	103	8	{	{	PUNCT
ap-4589	103	9	±	±	NOUN
ap-4589	103	10	1	1	NUM
ap-4589	103	11	2	2	NUM
ap-4589	103	12	,	,	PUNCT
ap-4589	103	13	±	±	NOUN
ap-4589	103	14	3	3	NUM
ap-4589	103	15	2	2	NUM
ap-4589	103	16	,	,	PUNCT
ap-4589	103	17	·	·	PUNCT
ap-4589	103	18	·	·	PUNCT
ap-4589	103	19	·	·	PUNCT
ap-4589	103	20	}	}	PUNCT
ap-4589	103	21	be	be	AUX
ap-4589	103	22	fixed	fix	VERB
ap-4589	103	23	.	.	PUNCT
ap-4589	104	1	since	since	SCONJ
ap-4589	104	2	the	the	DET
ap-4589	104	3	hamiltonian	hamiltonian	NOUN
ap-4589	104	4	is	be	AUX
ap-4589	104	5	invariant	invariant	ADJ
ap-4589	104	6	under	under	ADP
ap-4589	104	7	j	j	PROPN
ap-4589	104	8	→	→	SYM
ap-4589	104	9	1	1	NUM
ap-4589	104	10	−	−	PROPN
ap-4589	104	11	j	j	PROPN
ap-4589	104	12	,	,	PUNCT
ap-4589	104	13	without	without	ADP
ap-4589	104	14	any	any	DET
ap-4589	104	15	loss	loss	NOUN
ap-4589	104	16	of	of	ADP
ap-4589	104	17	generality	generality	NOUN
ap-4589	104	18	we	we	PRON
ap-4589	104	19	can	can	AUX
ap-4589	104	20	focus	focus	VERB
ap-4589	104	21	on	on	ADP
ap-4589	104	22	the	the	DET
ap-4589	104	23	case	case	NOUN
ap-4589	104	24	j	j	PROPN
ap-4589	104	25	∈	∈	PROPN
ap-4589	104	26	{	{	PUNCT
ap-4589	104	27	1	1	NUM
ap-4589	104	28	2	2	NUM
ap-4589	104	29	,	,	PUNCT
ap-4589	104	30	3	3	NUM
ap-4589	104	31	2	2	NUM
ap-4589	104	32	,	,	PUNCT
ap-4589	104	33	·	·	PUNCT
ap-4589	104	34	·	·	PUNCT
ap-4589	104	35	·	·	PUNCT
ap-4589	104	36	}	}	PUNCT
ap-4589	104	37	.	.	PUNCT
ap-4589	105	1	then	then	ADV
ap-4589	105	2	the	the	DET
ap-4589	105	3	discrete	discrete	ADJ
ap-4589	105	4	energy	energy	NOUN
ap-4589	105	5	eigenvalues	eigenvalue	NOUN
ap-4589	105	6	read	read	VERB
ap-4589	105	7	en	en	ADV
ap-4589	105	8	=	=	SYM
ap-4589	105	9	−	−	PROPN
ap-4589	105	10	g2	g2	PROPN
ap-4589	105	11	(	(	PUNCT
ap-4589	105	12	j	j	PROPN
ap-4589	105	13	+	+	PROPN
ap-4589	105	14	n)2	n)2	PROPN
ap-4589	105	15	,	,	PUNCT
ap-4589	105	16	n	n	CCONJ
ap-4589	105	17	∈	∈	PROPN
ap-4589	105	18	{	{	PUNCT
ap-4589	105	19	0	0	NUM
ap-4589	105	20	,	,	PUNCT
ap-4589	105	21	1	1	NUM
ap-4589	105	22	,	,	PUNCT
ap-4589	105	23	·	·	PUNCT
ap-4589	105	24	·	·	PUNCT
ap-4589	105	25	·	·	PUNCT
ap-4589	105	26	}	}	PUNCT
ap-4589	105	27	.	.	PUNCT
ap-4589	106	1	(	(	PUNCT
ap-4589	106	2	2.10	2.10	NUM
ap-4589	106	3	)	)	PUNCT
ap-4589	106	4	the	the	DET
ap-4589	106	5	energy	energy	NOUN
ap-4589	106	6	eigenfunction	eigenfunction	NOUN
ap-4589	106	7	ψen	ψen	NOUN
ap-4589	106	8	,	,	PUNCT
ap-4589	106	9	j(x	j(x	PROPN
ap-4589	106	10	)	)	PUNCT
ap-4589	106	11	that	that	PRON
ap-4589	106	12	satisfies	satisfy	VERB
ap-4589	106	13	the	the	DET
ap-4589	106	14	schrödinger	schrödinger	ADJ
ap-4589	106	15	equation	equation	NOUN
ap-4589	106	16	hkeplerψen	hkeplerψen	NOUN
ap-4589	106	17	,	,	PUNCT
ap-4589	106	18	j	j	PROPN
ap-4589	106	19	=	=	PUNCT
ap-4589	106	20	enψen	enψen	PROPN
ap-4589	106	21	,	,	PUNCT
ap-4589	106	22	j	j	PROPN
ap-4589	106	23	can	can	AUX
ap-4589	106	24	be	be	AUX
ap-4589	106	25	determined	determine	VERB
ap-4589	106	26	by	by	ADP
ap-4589	106	27	the	the	DET
ap-4589	106	28	formula	formula	NOUN
ap-4589	106	29	|en	|en	ADP
ap-4589	106	30	,	,	PUNCT
ap-4589	106	31	j	j	PROPN
ap-4589	106	32	〉	〉	PROPN
ap-4589	106	33	∝	∝	PROPN
ap-4589	106	34	(	(	PUNCT
ap-4589	106	35	j−)n|en	j−)n|en	PROPN
ap-4589	106	36	,	,	PUNCT
ap-4589	106	37	j+n	j+n	PROPN
ap-4589	106	38	〉	〉	NOUN
ap-4589	106	39	.	.	PUNCT
ap-4589	107	1	noting	note	VERB
ap-4589	107	2	that	that	SCONJ
ap-4589	107	3	|e	|e	PROPN
ap-4589	107	4	,	,	PUNCT
ap-4589	107	5	j	j	PROPN
ap-4589	107	6	〉	〉	PROPN
ap-4589	107	7	corresponds	correspond	VERB
ap-4589	107	8	to	to	ADP
ap-4589	107	9	the	the	DET
ap-4589	107	10	function	function	NOUN
ap-4589	107	11	ψe	ψe	NOUN
ap-4589	107	12	,	,	PUNCT
ap-4589	107	13	j(x)eijθ	j(x)eijθ	NOUN
ap-4589	107	14	and	and	CCONJ
ap-4589	107	15	j−	j−	PROPN
ap-4589	107	16	is	be	AUX
ap-4589	107	17	given	give	VERB
ap-4589	107	18	by	by	ADP
ap-4589	107	19	j−	j−	PROPN
ap-4589	107	20	=	=	SYM
ap-4589	107	21	a−(j3)e−iθ	a−(j3)e−iθ	PROPN
ap-4589	107	22	,	,	PUNCT
ap-4589	107	23	we	we	PRON
ap-4589	107	24	get	get	VERB
ap-4589	107	25	the	the	DET
ap-4589	107	26	following	follow	VERB
ap-4589	107	27	rodrigues	rodrigue	NOUN
ap-4589	107	28	-	-	PUNCT
ap-4589	107	29	like	like	ADJ
ap-4589	107	30	formula	formula	NOUN
ap-4589	107	31	:	:	PUNCT
ap-4589	107	32	ψen	ψen	NOUN
ap-4589	107	33	,	,	PUNCT
ap-4589	107	34	j(x	j(x	PROPN
ap-4589	107	35	)	)	PUNCT
ap-4589	107	36	∝	∝	X
ap-4589	108	1	a−(j)a−(j	a−(j)a−(j	PRON
ap-4589	108	2	+	+	NUM
ap-4589	108	3	1	1	NUM
ap-4589	108	4	)	)	PUNCT
ap-4589	108	5	·	·	PUNCT
ap-4589	108	6	·	·	PUNCT
ap-4589	109	1	·	·	PUNCT
ap-4589	109	2	a−(j	a−(j	PROPN
ap-4589	109	3	+	+	CCONJ
ap-4589	109	4	n−	n−	NOUN
ap-4589	109	5	1)ψen	1)ψen	NUM
ap-4589	109	6	,	,	PUNCT
ap-4589	109	7	j+n(x	j+n(x	NOUN
ap-4589	109	8	)	)	PUNCT
ap-4589	109	9	,	,	PUNCT
ap-4589	109	10	(	(	PUNCT
ap-4589	109	11	2.11	2.11	NUM
ap-4589	109	12	)	)	PUNCT
ap-4589	109	13	where	where	SCONJ
ap-4589	109	14	ψen	ψen	NOUN
ap-4589	109	15	,	,	PUNCT
ap-4589	109	16	j+n(x	j+n(x	NOUN
ap-4589	109	17	)	)	PUNCT
ap-4589	109	18	is	be	AUX
ap-4589	109	19	a	a	DET
ap-4589	109	20	solution	solution	NOUN
ap-4589	109	21	to	to	ADP
ap-4589	109	22	the	the	DET
ap-4589	109	23	first	first	ADJ
ap-4589	109	24	-	-	PUNCT
ap-4589	109	25	order	order	NOUN
ap-4589	109	26	differential	differential	ADJ
ap-4589	109	27	equation	equation	NOUN
ap-4589	109	28	a+(j	a+(j	PROPN
ap-4589	109	29	+	+	CCONJ
ap-4589	109	30	n)ψen	n)ψen	NOUN
ap-4589	109	31	,	,	PUNCT
ap-4589	109	32	j+n(x	j+n(x	NOUN
ap-4589	109	33	)	)	PUNCT
ap-4589	109	34	=	=	SYM
ap-4589	109	35	0	0	PUNCT
ap-4589	109	36	and	and	CCONJ
ap-4589	109	37	given	give	VERB
ap-4589	109	38	by	by	ADP
ap-4589	109	39	ψen	ψen	NOUN
ap-4589	109	40	,	,	PUNCT
ap-4589	109	41	j+n(x	j+n(x	NOUN
ap-4589	109	42	)	)	PUNCT
ap-4589	109	43	∝	∝	PROPN
ap-4589	109	44	xj+n	xj+n	PROPN
ap-4589	109	45	exp(−	exp(−	PROPN
ap-4589	109	46	g	g	PROPN
ap-4589	109	47	j+nx	j+nx	VERB
ap-4589	109	48	)	)	PUNCT
ap-4589	109	49	.	.	PUNCT
ap-4589	110	1	all	all	PRON
ap-4589	110	2	of	of	ADP
ap-4589	110	3	these	these	DET
ap-4589	110	4	exactly	exactly	ADV
ap-4589	110	5	coincide	coincide	NOUN
ap-4589	110	6	with	with	ADP
ap-4589	110	7	the	the	DET
ap-4589	110	8	well	well	ADV
ap-4589	110	9	-	-	PUNCT
ap-4589	110	10	known	know	VERB
ap-4589	110	11	results	result	NOUN
ap-4589	110	12	.	.	PUNCT
ap-4589	111	1	in	in	ADP
ap-4589	111	2	the	the	DET
ap-4589	111	3	rest	rest	NOUN
ap-4589	111	4	of	of	ADP
ap-4589	111	5	the	the	DET
ap-4589	111	6	note	note	NOUN
ap-4589	111	7	we	we	PRON
ap-4589	111	8	would	would	AUX
ap-4589	111	9	like	like	VERB
ap-4589	111	10	to	to	PART
ap-4589	111	11	apply	apply	VERB
ap-4589	111	12	the	the	DET
ap-4589	111	13	same	same	ADJ
ap-4589	111	14	idea	idea	NOUN
ap-4589	111	15	to	to	ADP
ap-4589	111	16	the	the	DET
ap-4589	111	17	spectral	spectral	ADJ
ap-4589	111	18	problem	problem	NOUN
ap-4589	111	19	for	for	ADP
ap-4589	111	20	the	the	DET
ap-4589	111	21	spherical	spherical	ADJ
ap-4589	111	22	kepler	kepler	NOUN
ap-4589	111	23	,	,	PUNCT
ap-4589	111	24	hyperbolic	hyperbolic	ADJ
ap-4589	111	25	kepler	kepler	NOUN
ap-4589	111	26	,	,	PUNCT
ap-4589	111	27	and	and	CCONJ
ap-4589	111	28	rosen	rosen	PROPN
ap-4589	111	29	–	–	PUNCT
ap-4589	111	30	morse	morse	PROPN
ap-4589	111	31	hamiltonians	hamiltonian	NOUN
ap-4589	111	32	.	.	PUNCT
ap-4589	112	1	we	we	PRON
ap-4589	112	2	shall	shall	AUX
ap-4589	112	3	first	first	ADV
ap-4589	112	4	introduce	introduce	VERB
ap-4589	112	5	the	the	DET
ap-4589	112	6	potential	potential	ADJ
ap-4589	112	7	algebras	algebra	NOUN
ap-4589	112	8	,	,	PUNCT
ap-4589	112	9	and	and	CCONJ
ap-4589	112	10	then	then	ADV
ap-4589	112	11	classify	classify	VERB
ap-4589	112	12	their	their	PRON
ap-4589	112	13	representations	representation	NOUN
ap-4589	112	14	,	,	PUNCT
ap-4589	112	15	and	and	CCONJ
ap-4589	112	16	then	then	ADV
ap-4589	112	17	solve	solve	VERB
ap-4589	112	18	the	the	DET
ap-4589	112	19	bound	bind	VERB
ap-4589	112	20	-	-	PUNCT
ap-4589	112	21	state	state	NOUN
ap-4589	112	22	problems	problem	NOUN
ap-4589	112	23	.	.	PUNCT
ap-4589	113	1	as	as	SCONJ
ap-4589	113	2	we	we	PRON
ap-4589	113	3	will	will	AUX
ap-4589	113	4	see	see	VERB
ap-4589	113	5	below	below	ADV
ap-4589	113	6	,	,	PUNCT
ap-4589	113	7	the	the	DET
ap-4589	113	8	spherical	spherical	ADJ
ap-4589	113	9	kepler	kepler	NOUN
ap-4589	113	10	problem	problem	NOUN
ap-4589	113	11	is	be	AUX
ap-4589	113	12	rather	rather	ADV
ap-4589	113	13	straightforward	straightforward	ADJ
ap-4589	113	14	but	but	CCONJ
ap-4589	113	15	the	the	DET
ap-4589	113	16	hyperbolic	hyperbolic	ADJ
ap-4589	113	17	kepler	kepler	PROPN
ap-4589	113	18	and	and	CCONJ
ap-4589	113	19	rosen	rosen	PROPN
ap-4589	113	20	–	–	PUNCT
ap-4589	113	21	morse	morse	ADJ
ap-4589	113	22	potential	potential	ADJ
ap-4589	113	23	problems	problem	NOUN
ap-4589	113	24	are	be	AUX
ap-4589	113	25	more	more	ADV
ap-4589	113	26	intriguing	intriguing	ADJ
ap-4589	113	27	and	and	CCONJ
ap-4589	113	28	require	require	VERB
ap-4589	113	29	careful	careful	ADJ
ap-4589	113	30	analysis	analysis	NOUN
ap-4589	113	31	.	.	PUNCT
ap-4589	114	1	3	3	X
ap-4589	114	2	.	.	X
ap-4589	114	3	spherical	spherical	ADJ
ap-4589	114	4	kepler	kepler	NOUN
ap-4589	114	5	let	let	VERB
ap-4589	114	6	us	we	PRON
ap-4589	114	7	next	next	ADJ
ap-4589	114	8	move	move	NOUN
ap-4589	114	9	on	on	ADP
ap-4589	114	10	to	to	ADP
ap-4589	114	11	the	the	DET
ap-4589	114	12	spherical	spherical	ADJ
ap-4589	114	13	kepler	kepler	NOUN
ap-4589	114	14	problem	problem	NOUN
ap-4589	115	1	[	[	X
ap-4589	115	2	1–3	1–3	NOUN
ap-4589	115	3	]	]	X
ap-4589	115	4	,	,	PUNCT
ap-4589	115	5	whose	whose	DET
ap-4589	115	6	hamiltonian	hamiltonian	NOUN
ap-4589	115	7	is	be	AUX
ap-4589	115	8	factorized	factorize	VERB
ap-4589	115	9	as	as	SCONJ
ap-4589	115	10	follows	follow	VERB
ap-4589	115	11	:	:	PUNCT
ap-4589	115	12	hspherical	hspherical	ADJ
ap-4589	115	13	kepler	kepler	NOUN
ap-4589	115	14	=	=	PUNCT
ap-4589	115	15	a−(j)a+(j	a−(j)a+(j	ADV
ap-4589	115	16	)	)	PUNCT
ap-4589	115	17	+	+	CCONJ
ap-4589	115	18	j2	j2	PROPN
ap-4589	115	19	−	−	PROPN
ap-4589	115	20	g2	g2	PROPN
ap-4589	115	21	j2	j2	PROPN
ap-4589	115	22	,	,	PUNCT
ap-4589	115	23	(	(	PUNCT
ap-4589	115	24	3.1	3.1	NUM
ap-4589	115	25	)	)	PUNCT
ap-4589	115	26	449	449	NUM
ap-4589	115	27	satoshi	satoshi	PROPN
ap-4589	115	28	ohya	ohya	PROPN
ap-4589	115	29	acta	acta	PROPN
ap-4589	115	30	polytechnica	polytechnica	PROPN
ap-4589	115	31	where	where	SCONJ
ap-4589	115	32	a±(j	a±(j	PROPN
ap-4589	115	33	)	)	PUNCT
ap-4589	115	34	=	=	SYM
ap-4589	115	35	±	±	NUM
ap-4589	116	1	d	d	NOUN
ap-4589	116	2	dx	dx	PROPN
ap-4589	117	1	−	−	PROPN
ap-4589	117	2	j	j	PROPN
ap-4589	117	3	cotx+	cotx+	VERB
ap-4589	117	4	g	g	PROPN
ap-4589	117	5	j	j	PROPN
ap-4589	117	6	.	.	PUNCT
ap-4589	118	1	(	(	PUNCT
ap-4589	118	2	3.2	3.2	NUM
ap-4589	118	3	)	)	PUNCT
ap-4589	118	4	just	just	ADV
ap-4589	118	5	as	as	SCONJ
ap-4589	118	6	in	in	ADP
ap-4589	118	7	the	the	DET
ap-4589	118	8	previous	previous	ADJ
ap-4589	118	9	section	section	NOUN
ap-4589	118	10	,	,	PUNCT
ap-4589	118	11	let	let	VERB
ap-4589	118	12	us	we	PRON
ap-4589	118	13	next	next	ADJ
ap-4589	118	14	introduce	introduce	VERB
ap-4589	118	15	the	the	DET
ap-4589	118	16	following	follow	VERB
ap-4589	118	17	operators	operator	NOUN
ap-4589	118	18	:	:	PUNCT
ap-4589	118	19	j3	j3	PROPN
ap-4589	118	20	=	=	PUNCT
ap-4589	118	21	−i∂θ	−i∂θ	PROPN
ap-4589	118	22	,	,	PUNCT
ap-4589	118	23	j+	j+	NUM
ap-4589	119	1	=	=	SYM
ap-4589	119	2	eiθ	eiθ	PROPN
ap-4589	119	3	(	(	PUNCT
ap-4589	119	4	∂x	∂x	PROPN
ap-4589	119	5	−	−	PROPN
ap-4589	119	6	cotxj3	cotxj3	NOUN
ap-4589	119	7	+	+	CCONJ
ap-4589	119	8	g	g	PROPN
ap-4589	119	9	j3	j3	PROPN
ap-4589	119	10	)	)	PUNCT
ap-4589	119	11	,	,	PUNCT
ap-4589	119	12	j−	j−	PROPN
ap-4589	119	13	=	=	PUNCT
ap-4589	119	14	(	(	PUNCT
ap-4589	119	15	−∂x	−∂x	NUM
ap-4589	119	16	−	−	PROPN
ap-4589	119	17	cotxj3	cotxj3	NOUN
ap-4589	119	18	+	+	CCONJ
ap-4589	119	19	g	g	PROPN
ap-4589	119	20	j3	j3	PROPN
ap-4589	119	21	)	)	PUNCT
ap-4589	119	22	e−iθ	e−iθ	PROPN
ap-4589	119	23	,	,	PUNCT
ap-4589	119	24	(	(	PUNCT
ap-4589	119	25	3.3	3.3	NUM
ap-4589	119	26	)	)	PUNCT
ap-4589	119	27	which	which	PRON
ap-4589	119	28	satisfy	satisfy	VERB
ap-4589	119	29	the	the	DET
ap-4589	119	30	following	follow	VERB
ap-4589	119	31	commutation	commutation	NOUN
ap-4589	119	32	relations	relation	NOUN
ap-4589	119	33	:	:	PUNCT
ap-4589	120	1	[	[	X
ap-4589	120	2	j3	j3	PROPN
ap-4589	120	3	,	,	PUNCT
ap-4589	120	4	j±	j±	PROPN
ap-4589	120	5	]	]	X
ap-4589	120	6	=	=	SYM
ap-4589	120	7	±j±	±j±	PROPN
ap-4589	120	8	,	,	PUNCT
ap-4589	120	9	[	[	X
ap-4589	120	10	j+	j+	NUM
ap-4589	120	11	,	,	PUNCT
ap-4589	120	12	j−	j−	PROPN
ap-4589	120	13	]	]	X
ap-4589	120	14	=	=	SYM
ap-4589	120	15	j2	j2	PROPN
ap-4589	120	16	3	3	NUM
ap-4589	120	17	−	−	PROPN
ap-4589	120	18	g2	g2	PROPN
ap-4589	120	19	j2	j2	PROPN
ap-4589	120	20	3	3	NUM
ap-4589	120	21	−	−	PROPN
ap-4589	120	22	(	(	PUNCT
ap-4589	120	23	j3	j3	PROPN
ap-4589	120	24	−	−	PROPN
ap-4589	120	25	1)2	1)2	PROPN
ap-4589	120	26	+	+	CCONJ
ap-4589	120	27	g2	g2	PROPN
ap-4589	120	28	(	(	PUNCT
ap-4589	120	29	j3	j3	PROPN
ap-4589	120	30	−	−	PROPN
ap-4589	120	31	1)2	1)2	PROPN
ap-4589	120	32	.	.	PUNCT
ap-4589	121	1	(	(	PUNCT
ap-4589	121	2	3.4	3.4	NUM
ap-4589	121	3	)	)	PUNCT
ap-4589	121	4	the	the	DET
ap-4589	121	5	invariant	invariant	ADJ
ap-4589	121	6	operator	operator	NOUN
ap-4589	121	7	that	that	PRON
ap-4589	121	8	commutes	commute	VERB
ap-4589	121	9	with	with	ADP
ap-4589	121	10	j3	j3	PROPN
ap-4589	121	11	and	and	CCONJ
ap-4589	121	12	j±	j±	PROPN
ap-4589	121	13	is	be	AUX
ap-4589	121	14	given	give	VERB
ap-4589	121	15	by	by	ADP
ap-4589	121	16	h	h	NOUN
ap-4589	121	17	=	=	PUNCT
ap-4589	121	18	j−j+	j−j+	PROPN
ap-4589	122	1	+	+	CCONJ
ap-4589	122	2	j2	j2	PROPN
ap-4589	122	3	3	3	NUM
ap-4589	122	4	−	−	PROPN
ap-4589	122	5	g2	g2	PROPN
ap-4589	122	6	j2	j2	PROPN
ap-4589	122	7	3	3	NUM
ap-4589	122	8	=	=	SYM
ap-4589	122	9	j+j−	j+j−	PROPN
ap-4589	122	10	+	+	CCONJ
ap-4589	122	11	(	(	PUNCT
ap-4589	122	12	j3	j3	PROPN
ap-4589	122	13	−	−	PROPN
ap-4589	122	14	1)2	1)2	PROPN
ap-4589	122	15	−	−	PROPN
ap-4589	122	16	g2	g2	PROPN
ap-4589	122	17	(	(	PUNCT
ap-4589	122	18	j3	j3	PROPN
ap-4589	122	19	−	−	PROPN
ap-4589	122	20	1)2	1)2	NUM
ap-4589	122	21	=	=	PUNCT
ap-4589	122	22	−∂2	−∂2	NOUN
ap-4589	122	23	x	x	PUNCT
ap-4589	123	1	+	+	PUNCT
ap-4589	123	2	j3(j3	j3(j3	NUM
ap-4589	123	3	−	−	PROPN
ap-4589	123	4	1	1	NUM
ap-4589	123	5	)	)	PUNCT
ap-4589	123	6	sin2	sin2	NOUN
ap-4589	123	7	x	x	PUNCT
ap-4589	124	1	−	−	PROPN
ap-4589	124	2	2	2	NUM
ap-4589	124	3	g	g	NOUN
ap-4589	124	4	cotx	cotx	NOUN
ap-4589	124	5	.	.	PUNCT
ap-4589	125	1	(	(	PUNCT
ap-4589	125	2	3.5	3.5	NUM
ap-4589	125	3	)	)	PUNCT
ap-4589	125	4	it	it	PRON
ap-4589	125	5	should	should	AUX
ap-4589	125	6	be	be	AUX
ap-4589	125	7	noted	note	VERB
ap-4589	125	8	that	that	SCONJ
ap-4589	125	9	,	,	PUNCT
ap-4589	125	10	if	if	SCONJ
ap-4589	125	11	g	g	PROPN
ap-4589	125	12	=	=	SYM
ap-4589	125	13	0	0	NUM
ap-4589	125	14	,	,	PUNCT
ap-4589	125	15	(	(	PUNCT
ap-4589	125	16	3.4	3.4	NUM
ap-4589	125	17	)	)	PUNCT
ap-4589	125	18	reduces	reduce	VERB
ap-4589	125	19	to	to	ADP
ap-4589	125	20	the	the	DET
ap-4589	125	21	standard	standard	ADJ
ap-4589	125	22	commutation	commutation	NOUN
ap-4589	125	23	relations	relation	NOUN
ap-4589	125	24	for	for	ADP
ap-4589	125	25	the	the	DET
ap-4589	125	26	lie	lie	NOUN
ap-4589	125	27	algebra	algebra	NOUN
ap-4589	125	28	so(3	so(3	NOUN
ap-4589	125	29	)	)	PUNCT
ap-4589	125	30	under	under	ADP
ap-4589	125	31	the	the	DET
ap-4589	125	32	appropriate	appropriate	ADJ
ap-4589	125	33	shift	shift	NOUN
ap-4589	125	34	j3	j3	PROPN
ap-4589	125	35	→	→	SYM
ap-4589	125	36	j3	j3	PROPN
ap-4589	125	37	+	+	PROPN
ap-4589	125	38	1/2	1/2	NUM
ap-4589	125	39	.	.	PUNCT
ap-4589	126	1	in	in	ADP
ap-4589	126	2	this	this	DET
ap-4589	126	3	case	case	NOUN
ap-4589	126	4	the	the	DET
ap-4589	126	5	invariant	invariant	ADJ
ap-4589	126	6	operator	operator	NOUN
ap-4589	126	7	h	h	NOUN
ap-4589	126	8	is	be	AUX
ap-4589	126	9	nothing	nothing	PRON
ap-4589	126	10	but	but	SCONJ
ap-4589	126	11	the	the	DET
ap-4589	126	12	casimir	casimir	NOUN
ap-4589	126	13	operator	operator	NOUN
ap-4589	126	14	of	of	ADP
ap-4589	126	15	so(3	so(3	NOUN
ap-4589	126	16	)	)	PUNCT
ap-4589	126	17	and	and	CCONJ
ap-4589	126	18	provides	provide	VERB
ap-4589	126	19	a	a	DET
ap-4589	126	20	well	well	ADV
ap-4589	126	21	-	-	PUNCT
ap-4589	126	22	known	know	VERB
ap-4589	126	23	example	example	NOUN
ap-4589	126	24	of	of	ADP
ap-4589	126	25	interplay	interplay	NOUN
ap-4589	126	26	between	between	ADP
ap-4589	126	27	shape	shape	NOUN
ap-4589	126	28	invariance	invariance	NOUN
ap-4589	126	29	and	and	CCONJ
ap-4589	126	30	lie	lie	NOUN
ap-4589	126	31	algebra	algebra	NOUN
ap-4589	126	32	;	;	PUNCT
ap-4589	126	33	see	see	VERB
ap-4589	126	34	,	,	PUNCT
ap-4589	126	35	e.g.	e.g.	ADV
ap-4589	126	36	,	,	PUNCT
ap-4589	126	37	the	the	DET
ap-4589	126	38	review	review	NOUN
ap-4589	126	39	[	[	X
ap-4589	126	40	12	12	NUM
ap-4589	126	41	]	]	PUNCT
ap-4589	126	42	.	.	PUNCT
ap-4589	127	1	now	now	ADV
ap-4589	127	2	,	,	PUNCT
ap-4589	127	3	let	let	VERB
ap-4589	127	4	|e	|e	PROPN
ap-4589	127	5	,	,	PUNCT
ap-4589	127	6	j	j	PROPN
ap-4589	127	7	〉	〉	PROPN
ap-4589	127	8	be	be	AUX
ap-4589	127	9	a	a	DET
ap-4589	127	10	simultaneous	simultaneous	ADJ
ap-4589	127	11	eigenstate	eigenstate	NOUN
ap-4589	127	12	of	of	ADP
ap-4589	127	13	h	h	PROPN
ap-4589	127	14	and	and	CCONJ
ap-4589	127	15	j3	j3	PROPN
ap-4589	127	16	that	that	PRON
ap-4589	127	17	satisfies	satisfy	VERB
ap-4589	127	18	the	the	DET
ap-4589	127	19	eigenvalue	eigenvalue	PROPN
ap-4589	127	20	equations	equation	NOUN
ap-4589	127	21	h|e	h|e	PROPN
ap-4589	127	22	,	,	PUNCT
ap-4589	127	23	j	j	PROPN
ap-4589	127	24	〉	〉	PROPN
ap-4589	127	25	=	=	SYM
ap-4589	127	26	e|e	e|e	PROPN
ap-4589	127	27	,	,	PUNCT
ap-4589	127	28	j	j	PROPN
ap-4589	127	29	〉	〉	PROPN
ap-4589	127	30	,	,	PUNCT
ap-4589	127	31	j3|e	j3|e	PROPN
ap-4589	127	32	,	,	PUNCT
ap-4589	127	33	j	j	PROPN
ap-4589	127	34	〉	〉	PROPN
ap-4589	127	35	=	=	SYM
ap-4589	127	36	j|e	j|e	PROPN
ap-4589	127	37	,	,	PUNCT
ap-4589	127	38	j	j	PROPN
ap-4589	127	39	〉	〉	PROPN
ap-4589	127	40	,	,	PUNCT
ap-4589	127	41	(	(	PUNCT
ap-4589	127	42	3.6	3.6	NUM
ap-4589	127	43	)	)	PUNCT
ap-4589	127	44	as	as	ADV
ap-4589	127	45	well	well	ADV
ap-4589	127	46	as	as	ADP
ap-4589	127	47	the	the	DET
ap-4589	127	48	normalization	normalization	NOUN
ap-4589	127	49	condition	condition	NOUN
ap-4589	127	50	‖|e	‖|e	PROPN
ap-4589	127	51	,	,	PUNCT
ap-4589	127	52	j〉‖	j〉‖	NOUN
ap-4589	127	53	=	=	SYM
ap-4589	128	1	1	1	X
ap-4589	128	2	.	.	PUNCT
ap-4589	128	3	then	then	ADV
ap-4589	128	4	we	we	PRON
ap-4589	128	5	have	have	VERB
ap-4589	128	6	the	the	DET
ap-4589	128	7	following	follow	VERB
ap-4589	128	8	conditions	condition	NOUN
ap-4589	128	9	:	:	PUNCT
ap-4589	128	10	‖j+|e	‖j+|e	PROPN
ap-4589	128	11	,	,	PUNCT
ap-4589	128	12	j〉‖2	j〉‖2	NOUN
ap-4589	128	13	=	=	SYM
ap-4589	129	1	e	e	X
ap-4589	129	2	−	−	PROPN
ap-4589	129	3	j2	j2	PROPN
ap-4589	129	4	+	+	CCONJ
ap-4589	129	5	g2	g2	PROPN
ap-4589	129	6	j2	j2	PROPN
ap-4589	129	7	≥	≥	PROPN
ap-4589	129	8	0	0	NUM
ap-4589	129	9	,	,	PUNCT
ap-4589	129	10	‖j−|e	‖j−|e	NUM
ap-4589	129	11	,	,	PUNCT
ap-4589	129	12	j〉‖2	j〉‖2	NOUN
ap-4589	129	13	=	=	PUNCT
ap-4589	129	14	e	e	X
ap-4589	129	15	−	−	PROPN
ap-4589	129	16	(	(	PUNCT
ap-4589	129	17	j	j	PROPN
ap-4589	129	18	−	−	PROPN
ap-4589	129	19	1)2	1)2	NUM
ap-4589	129	20	+	+	NUM
ap-4589	129	21	g2	g2	PROPN
ap-4589	129	22	(	(	PUNCT
ap-4589	129	23	j	j	PROPN
ap-4589	129	24	−	−	PROPN
ap-4589	129	25	1)2	1)2	NUM
ap-4589	129	26	≥	≥	NOUN
ap-4589	129	27	0	0	NUM
ap-4589	129	28	,	,	PUNCT
ap-4589	129	29	(	(	PUNCT
ap-4589	129	30	3.7	3.7	NUM
ap-4589	129	31	)	)	PUNCT
ap-4589	129	32	which	which	PRON
ap-4589	129	33	,	,	PUNCT
ap-4589	129	34	together	together	ADV
ap-4589	129	35	with	with	ADP
ap-4589	129	36	the	the	DET
ap-4589	129	37	ladder	ladder	NOUN
ap-4589	129	38	equations	equation	NOUN
ap-4589	129	39	j±|e	j±|e	PROPN
ap-4589	129	40	,	,	PUNCT
ap-4589	129	41	j	j	PROPN
ap-4589	129	42	〉	〉	PROPN
ap-4589	129	43	∝	∝	PROPN
ap-4589	129	44	|e	|e	PROPN
ap-4589	129	45	,	,	PUNCT
ap-4589	129	46	j	j	PROPN
ap-4589	129	47	±	±	PROPN
ap-4589	129	48	1	1	NUM
ap-4589	129	49	〉	〉	NUM
ap-4589	129	50	,	,	PUNCT
ap-4589	129	51	restrict	restrict	VERB
ap-4589	129	52	the	the	DET
ap-4589	129	53	possible	possible	ADJ
ap-4589	129	54	values	value	NOUN
ap-4589	129	55	of	of	ADP
ap-4589	129	56	e	e	PROPN
ap-4589	129	57	and	and	CCONJ
ap-4589	129	58	j.	j.	PROPN
ap-4589	129	59	as	as	SCONJ
ap-4589	129	60	discussed	discuss	VERB
ap-4589	129	61	in	in	ADP
ap-4589	129	62	the	the	DET
ap-4589	129	63	previous	previous	ADJ
ap-4589	129	64	section	section	NOUN
ap-4589	129	65	,	,	PUNCT
ap-4589	129	66	these	these	DET
ap-4589	129	67	conditions	condition	NOUN
ap-4589	129	68	are	be	AUX
ap-4589	129	69	compatible	compatible	ADJ
ap-4589	129	70	with	with	ADP
ap-4589	129	71	each	each	DET
ap-4589	129	72	other	other	ADJ
ap-4589	129	73	if	if	SCONJ
ap-4589	129	74	and	and	CCONJ
ap-4589	129	75	only	only	ADV
ap-4589	129	76	if	if	SCONJ
ap-4589	129	77	the	the	DET
ap-4589	129	78	eigenvalue	eigenvalue	NOUN
ap-4589	129	79	of	of	ADP
ap-4589	129	80	the	the	DET
ap-4589	129	81	invariant	invariant	ADJ
ap-4589	129	82	operator	operator	NOUN
ap-4589	129	83	takes	take	VERB
ap-4589	129	84	the	the	DET
ap-4589	129	85	value	value	NOUN
ap-4589	129	86	e	e	NOUN
ap-4589	129	87	=	=	SYM
ap-4589	129	88	ν2	ν2	PROPN
ap-4589	129	89	−	−	PROPN
ap-4589	129	90	g2	g2	PROPN
ap-4589	129	91	/	/	SYM
ap-4589	129	92	ν2	ν2	PROPN
ap-4589	129	93	,	,	PUNCT
ap-4589	129	94	ν	ν	X
ap-4589	129	95	∈	∈	PROPN
ap-4589	129	96	{	{	PUNCT
ap-4589	129	97	1	1	NUM
ap-4589	129	98	2	2	NUM
ap-4589	129	99	,	,	PUNCT
ap-4589	129	100	3	3	NUM
ap-4589	129	101	2	2	NUM
ap-4589	129	102	,	,	PUNCT
ap-4589	129	103	·	·	PUNCT
ap-4589	129	104	·	·	PUNCT
ap-4589	129	105	·	·	PUNCT
ap-4589	130	1	}	}	PUNCT
ap-4589	130	2	.	.	PUNCT
ap-4589	131	1	now	now	ADV
ap-4589	131	2	let	let	VERB
ap-4589	131	3	ν	ν	X
ap-4589	131	4	∈	∈	PROPN
ap-4589	131	5	{	{	PUNCT
ap-4589	131	6	1	1	NUM
ap-4589	131	7	2	2	NUM
ap-4589	131	8	,	,	PUNCT
ap-4589	131	9	3	3	NUM
ap-4589	131	10	2	2	NUM
ap-4589	131	11	,	,	PUNCT
ap-4589	131	12	·	·	PUNCT
ap-4589	131	13	·	·	PUNCT
ap-4589	131	14	·	·	PUNCT
ap-4589	131	15	}	}	PUNCT
ap-4589	131	16	be	be	AUX
ap-4589	131	17	fixed	fix	VERB
ap-4589	131	18	.	.	PUNCT
ap-4589	132	1	then	then	ADV
ap-4589	132	2	the	the	DET
ap-4589	132	3	representation	representation	NOUN
ap-4589	132	4	space	space	NOUN
ap-4589	132	5	is	be	AUX
ap-4589	132	6	spanned	span	VERB
ap-4589	132	7	by	by	ADP
ap-4589	132	8	the	the	DET
ap-4589	132	9	following	follow	VERB
ap-4589	132	10	2ν	2ν	NUM
ap-4589	132	11	vectors	vector	NOUN
ap-4589	132	12	:	:	PUNCT
ap-4589	132	13	{	{	PUNCT
ap-4589	132	14	|e	|e	PROPN
ap-4589	132	15	,	,	PUNCT
ap-4589	132	16	j	j	PROPN
ap-4589	132	17	〉	〉	NOUN
ap-4589	132	18	:	:	PUNCT
ap-4589	132	19	e	e	X
ap-4589	132	20	=	=	PUNCT
ap-4589	132	21	ν2	ν2	PROPN
ap-4589	132	22	−	−	PROPN
ap-4589	132	23	g2	g2	PROPN
ap-4589	132	24	ν2	ν2	NOUN
ap-4589	132	25	and	and	CCONJ
ap-4589	132	26	j	j	PROPN
ap-4589	132	27	∈	∈	PROPN
ap-4589	132	28	{	{	PUNCT
ap-4589	132	29	ν	ν	NOUN
ap-4589	132	30	,	,	PUNCT
ap-4589	132	31	ν	ν	X
ap-4589	132	32	−	−	PROPN
ap-4589	132	33	1	1	NUM
ap-4589	132	34	,	,	PUNCT
ap-4589	132	35	·	·	PUNCT
ap-4589	132	36	·	·	PUNCT
ap-4589	132	37	·	·	PUNCT
ap-4589	132	38	,	,	PUNCT
ap-4589	132	39	1−	1−	NUM
ap-4589	132	40	ν	ν	X
ap-4589	132	41	}	}	PUNCT
ap-4589	132	42	}	}	PUNCT
ap-4589	132	43	.	.	PUNCT
ap-4589	133	1	(	(	PUNCT
ap-4589	133	2	3.8	3.8	NUM
ap-4589	133	3	)	)	PUNCT
ap-4589	133	4	these	these	DET
ap-4589	133	5	2ν	2ν	ADV
ap-4589	133	6	-	-	PUNCT
ap-4589	133	7	dimensional	dimensional	ADJ
ap-4589	133	8	representations	representation	NOUN
ap-4589	133	9	are	be	AUX
ap-4589	133	10	schematically	schematically	ADV
ap-4589	133	11	depicted	depict	VERB
ap-4589	133	12	in	in	ADP
ap-4589	133	13	figure	figure	NOUN
ap-4589	133	14	2(b	2(b	NUM
ap-4589	133	15	)	)	PUNCT
ap-4589	133	16	.	.	PUNCT
ap-4589	134	1	now	now	ADV
ap-4589	134	2	it	it	PRON
ap-4589	134	3	is	be	AUX
ap-4589	134	4	easy	easy	ADJ
ap-4589	134	5	to	to	PART
ap-4589	134	6	find	find	VERB
ap-4589	134	7	the	the	DET
ap-4589	134	8	spectrum	spectrum	NOUN
ap-4589	134	9	of	of	ADP
ap-4589	134	10	the	the	DET
ap-4589	134	11	original	original	ADJ
ap-4589	134	12	hamiltonian	hamiltonian	ADJ
ap-4589	134	13	hspherical	hspherical	ADJ
ap-4589	134	14	kepler	kepler	NOUN
ap-4589	134	15	.	.	PUNCT
ap-4589	135	1	for	for	ADP
ap-4589	135	2	fixed	fix	VERB
ap-4589	135	3	j	j	PROPN
ap-4589	135	4	∈	∈	PROPN
ap-4589	135	5	{	{	PUNCT
ap-4589	135	6	1	1	NUM
ap-4589	135	7	2	2	NUM
ap-4589	135	8	,	,	PUNCT
ap-4589	135	9	3	3	NUM
ap-4589	135	10	2	2	NUM
ap-4589	135	11	,	,	PUNCT
ap-4589	135	12	·	·	PUNCT
ap-4589	135	13	·	·	PUNCT
ap-4589	135	14	·	·	PUNCT
ap-4589	135	15	}	}	PUNCT
ap-4589	135	16	the	the	DET
ap-4589	135	17	energy	energy	NOUN
ap-4589	135	18	eigenvalues	eigenvalue	NOUN
ap-4589	135	19	and	and	CCONJ
ap-4589	135	20	eigenfunctions	eigenfunction	NOUN
ap-4589	135	21	read	read	VERB
ap-4589	135	22	en	en	ADV
ap-4589	135	23	=	=	SYM
ap-4589	135	24	(	(	PUNCT
ap-4589	135	25	j	j	PROPN
ap-4589	135	26	+	+	PROPN
ap-4589	135	27	n)2	n)2	PROPN
ap-4589	135	28	−	−	PROPN
ap-4589	135	29	g2	g2	PROPN
ap-4589	135	30	(	(	PUNCT
ap-4589	135	31	j	j	PROPN
ap-4589	135	32	+	+	PROPN
ap-4589	135	33	n)2	n)2	PROPN
ap-4589	135	34	,	,	PUNCT
ap-4589	135	35	n	n	CCONJ
ap-4589	135	36	∈	∈	PROPN
ap-4589	135	37	{	{	PUNCT
ap-4589	135	38	0	0	NUM
ap-4589	135	39	,	,	PUNCT
ap-4589	135	40	1	1	NUM
ap-4589	135	41	,	,	PUNCT
ap-4589	135	42	·	·	PUNCT
ap-4589	135	43	·	·	PUNCT
ap-4589	135	44	·	·	PUNCT
ap-4589	135	45	}	}	PUNCT
ap-4589	135	46	,	,	PUNCT
ap-4589	135	47	(	(	PUNCT
ap-4589	135	48	3.9	3.9	NUM
ap-4589	135	49	)	)	PUNCT
ap-4589	135	50	and	and	CCONJ
ap-4589	135	51	ψen	ψen	NOUN
ap-4589	135	52	,	,	PUNCT
ap-4589	135	53	j(x	j(x	PROPN
ap-4589	135	54	)	)	PUNCT
ap-4589	135	55	∝	∝	X
ap-4589	136	1	a−(j)a−(j	a−(j)a−(j	PRON
ap-4589	136	2	+	+	NUM
ap-4589	136	3	1	1	NUM
ap-4589	136	4	)	)	PUNCT
ap-4589	136	5	·	·	PUNCT
ap-4589	136	6	·	·	PUNCT
ap-4589	137	1	·	·	PUNCT
ap-4589	137	2	a−(j	a−(j	PROPN
ap-4589	137	3	+	+	CCONJ
ap-4589	137	4	n−	n−	NOUN
ap-4589	137	5	1)ψen	1)ψen	NUM
ap-4589	137	6	,	,	PUNCT
ap-4589	137	7	j+n(x	j+n(x	NOUN
ap-4589	137	8	)	)	PUNCT
ap-4589	137	9	,	,	PUNCT
ap-4589	137	10	(	(	PUNCT
ap-4589	137	11	3.10	3.10	NUM
ap-4589	137	12	)	)	PUNCT
ap-4589	137	13	where	where	SCONJ
ap-4589	137	14	ψen	ψen	NOUN
ap-4589	137	15	,	,	PUNCT
ap-4589	137	16	j+n(x	j+n(x	NOUN
ap-4589	137	17	)	)	PUNCT
ap-4589	137	18	∝	∝	PROPN
ap-4589	137	19	(	(	PUNCT
ap-4589	137	20	sin	sin	PROPN
ap-4589	137	21	x)j+n	x)j+n	PROPN
ap-4589	137	22	exp(−	exp(−	PROPN
ap-4589	137	23	g	g	PROPN
ap-4589	137	24	j+nx	j+nx	VERB
ap-4589	137	25	)	)	PUNCT
ap-4589	137	26	.	.	PUNCT
ap-4589	138	1	we	we	PRON
ap-4589	138	2	note	note	VERB
ap-4589	138	3	that	that	SCONJ
ap-4589	138	4	(	(	PUNCT
ap-4589	138	5	3.9	3.9	NUM
ap-4589	138	6	)	)	PUNCT
ap-4589	138	7	and	and	CCONJ
ap-4589	138	8	(	(	PUNCT
ap-4589	138	9	3.10	3.10	NUM
ap-4589	138	10	)	)	PUNCT
ap-4589	138	11	are	be	AUX
ap-4589	138	12	consistent	consistent	ADJ
ap-4589	138	13	with	with	ADP
ap-4589	138	14	the	the	DET
ap-4589	138	15	known	know	VERB
ap-4589	138	16	results	result	NOUN
ap-4589	138	17	[	[	X
ap-4589	138	18	1–3	1–3	NOUN
ap-4589	138	19	]	]	X
ap-4589	138	20	.	.	PUNCT
ap-4589	139	1	4	4	X
ap-4589	139	2	.	.	X
ap-4589	139	3	hyperbolic	hyperbolic	PROPN
ap-4589	139	4	kepler	kepler	PROPN
ap-4589	139	5	&	&	CCONJ
ap-4589	139	6	rosen	rosen	PROPN
ap-4589	139	7	–	–	PUNCT
ap-4589	139	8	morse	morse	PROPN
ap-4589	139	9	let	let	VERB
ap-4589	139	10	us	we	PRON
ap-4589	139	11	finally	finally	ADV
ap-4589	139	12	move	move	VERB
ap-4589	139	13	on	on	ADP
ap-4589	139	14	to	to	ADP
ap-4589	139	15	the	the	DET
ap-4589	139	16	study	study	NOUN
ap-4589	139	17	of	of	ADP
ap-4589	139	18	potential	potential	ADJ
ap-4589	139	19	algebras	algebra	NOUN
ap-4589	139	20	for	for	ADP
ap-4589	139	21	the	the	DET
ap-4589	139	22	hyperbolic	hyperbolic	ADJ
ap-4589	139	23	kepler	kepler	NOUN
ap-4589	139	24	and	and	CCONJ
ap-4589	139	25	rosen	rosen	PROPN
ap-4589	139	26	–	–	PUNCT
ap-4589	139	27	morse	morse	NOUN
ap-4589	139	28	hamiltonians	hamiltonian	NOUN
ap-4589	139	29	.	.	PUNCT
ap-4589	140	1	we	we	PRON
ap-4589	140	2	shall	shall	AUX
ap-4589	140	3	see	see	VERB
ap-4589	140	4	that	that	SCONJ
ap-4589	140	5	the	the	DET
ap-4589	140	6	bound	bind	VERB
ap-4589	140	7	-	-	PUNCT
ap-4589	140	8	state	state	NOUN
ap-4589	140	9	spectra	spectra	NOUN
ap-4589	140	10	of	of	ADP
ap-4589	140	11	these	these	DET
ap-4589	140	12	problems	problem	NOUN
ap-4589	140	13	correspond	correspond	VERB
ap-4589	140	14	to	to	ADP
ap-4589	140	15	two	two	NUM
ap-4589	140	16	distinct	distinct	ADJ
ap-4589	140	17	representations	representation	NOUN
ap-4589	140	18	of	of	ADP
ap-4589	140	19	a	a	DET
ap-4589	140	20	single	single	ADJ
ap-4589	140	21	algebraic	algebraic	ADJ
ap-4589	140	22	system	system	NOUN
ap-4589	140	23	.	.	PUNCT
ap-4589	141	1	4.1	4.1	NUM
ap-4589	141	2	.	.	PUNCT
ap-4589	141	3	hyperbolic	hyperbolic	ADJ
ap-4589	141	4	kepler	kepler	PROPN
ap-4589	141	5	the	the	DET
ap-4589	141	6	hamiltonian	hamiltonian	NOUN
ap-4589	141	7	for	for	ADP
ap-4589	141	8	the	the	DET
ap-4589	141	9	hyperbolic	hyperbolic	ADJ
ap-4589	141	10	kepler	kepler	NOUN
ap-4589	141	11	problem	problem	NOUN
ap-4589	142	1	[	[	X
ap-4589	142	2	4	4	NUM
ap-4589	142	3	,	,	PUNCT
ap-4589	142	4	5	5	NUM
ap-4589	142	5	]	]	PUNCT
ap-4589	142	6	can	can	AUX
ap-4589	142	7	be	be	AUX
ap-4589	142	8	factorized	factorize	VERB
ap-4589	142	9	as	as	SCONJ
ap-4589	142	10	follows	follow	VERB
ap-4589	142	11	:	:	PUNCT
ap-4589	142	12	hhyperbolic	hhyperbolic	ADJ
ap-4589	142	13	kepler	kepler	NOUN
ap-4589	142	14	=	=	PUNCT
ap-4589	142	15	a−(j)a+(j)−	a−(j)a+(j)−	PROPN
ap-4589	142	16	j2	j2	PROPN
ap-4589	142	17	−	−	PROPN
ap-4589	142	18	g2	g2	PROPN
ap-4589	142	19	j2	j2	PROPN
ap-4589	142	20	,	,	PUNCT
ap-4589	142	21	(	(	PUNCT
ap-4589	142	22	4.1	4.1	NUM
ap-4589	142	23	)	)	PUNCT
ap-4589	142	24	450	450	NUM
ap-4589	142	25	vol	vol	NOUN
ap-4589	142	26	.	.	PUNCT
ap-4589	143	1	57	57	NUM
ap-4589	143	2	no	no	NOUN
ap-4589	143	3	.	.	PUNCT
ap-4589	144	1	6/2017	6/2017	X
ap-4589	144	2	algebraic	algebraic	ADJ
ap-4589	144	3	description	description	NOUN
ap-4589	144	4	of	of	ADP
ap-4589	144	5	shape	shape	NOUN
ap-4589	144	6	invariance	invariance	NOUN
ap-4589	144	7	revisited	revisit	VERB
ap-4589	144	8	where	where	SCONJ
ap-4589	144	9	a±(j	a±(j	PROPN
ap-4589	144	10	)	)	PUNCT
ap-4589	144	11	=	=	SYM
ap-4589	145	1	±	±	NUM
ap-4589	146	1	d	d	NOUN
ap-4589	146	2	dx	dx	PROPN
ap-4589	147	1	−	−	PROPN
ap-4589	147	2	j	j	PROPN
ap-4589	147	3	coth	coth	PROPN
ap-4589	147	4	x+	x+	PROPN
ap-4589	147	5	g	g	PROPN
ap-4589	147	6	j	j	PROPN
ap-4589	147	7	.	.	PUNCT
ap-4589	148	1	(	(	PUNCT
ap-4589	148	2	4.2	4.2	NUM
ap-4589	148	3	)	)	PUNCT
ap-4589	148	4	we	we	PRON
ap-4589	148	5	then	then	ADV
ap-4589	148	6	introduce	introduce	VERB
ap-4589	148	7	the	the	DET
ap-4589	148	8	following	follow	VERB
ap-4589	148	9	operators	operator	NOUN
ap-4589	148	10	:	:	PUNCT
ap-4589	148	11	j3	j3	PROPN
ap-4589	148	12	=	=	PUNCT
ap-4589	148	13	−i∂θ	−i∂θ	PROPN
ap-4589	148	14	,	,	PUNCT
ap-4589	148	15	j+	j+	NUM
ap-4589	148	16	=	=	SYM
ap-4589	148	17	eiθ	eiθ	PROPN
ap-4589	148	18	(	(	PUNCT
ap-4589	148	19	∂x	∂x	PROPN
ap-4589	148	20	−	−	PROPN
ap-4589	148	21	coth	coth	NOUN
ap-4589	148	22	xj3	xj3	PROPN
ap-4589	149	1	+	+	CCONJ
ap-4589	149	2	g	g	PROPN
ap-4589	149	3	j3	j3	PROPN
ap-4589	149	4	)	)	PUNCT
ap-4589	149	5	,	,	PUNCT
ap-4589	149	6	j−	j−	PROPN
ap-4589	149	7	=	=	PUNCT
ap-4589	149	8	(	(	PUNCT
ap-4589	149	9	−∂x	−∂x	NUM
ap-4589	149	10	−	−	PROPN
ap-4589	149	11	coth	coth	PROPN
ap-4589	149	12	xj3	xj3	PROPN
ap-4589	150	1	+	+	CCONJ
ap-4589	150	2	g	g	PROPN
ap-4589	150	3	j3	j3	PROPN
ap-4589	150	4	)	)	PUNCT
ap-4589	150	5	e−iθ	e−iθ	PROPN
ap-4589	150	6	,	,	PUNCT
ap-4589	150	7	(	(	PUNCT
ap-4589	150	8	4.3	4.3	NUM
ap-4589	150	9	)	)	PUNCT
ap-4589	150	10	which	which	PRON
ap-4589	150	11	satisfy	satisfy	VERB
ap-4589	150	12	the	the	DET
ap-4589	150	13	following	follow	VERB
ap-4589	150	14	commutation	commutation	NOUN
ap-4589	150	15	relations	relation	NOUN
ap-4589	150	16	:	:	PUNCT
ap-4589	151	1	[	[	X
ap-4589	151	2	j3	j3	PROPN
ap-4589	151	3	,	,	PUNCT
ap-4589	151	4	j±	j±	PROPN
ap-4589	151	5	]	]	X
ap-4589	151	6	=	=	SYM
ap-4589	151	7	±j±	±j±	PROPN
ap-4589	151	8	,	,	PUNCT
ap-4589	151	9	[	[	X
ap-4589	151	10	j+	j+	NUM
ap-4589	151	11	,	,	PUNCT
ap-4589	151	12	j−	j−	PROPN
ap-4589	151	13	]	]	X
ap-4589	151	14	=	=	SYM
ap-4589	151	15	−j2	−j2	NOUN
ap-4589	151	16	3	3	NUM
ap-4589	151	17	−	−	PROPN
ap-4589	151	18	g2	g2	PROPN
ap-4589	151	19	j2	j2	PROPN
ap-4589	151	20	3	3	NUM
ap-4589	151	21	+	+	CCONJ
ap-4589	151	22	(	(	PUNCT
ap-4589	151	23	j3	j3	PROPN
ap-4589	151	24	−	−	PROPN
ap-4589	151	25	1)2	1)2	PROPN
ap-4589	151	26	+	+	CCONJ
ap-4589	151	27	g2	g2	PROPN
ap-4589	151	28	(	(	PUNCT
ap-4589	151	29	j3	j3	PROPN
ap-4589	151	30	−	−	PROPN
ap-4589	151	31	1)2	1)2	PROPN
ap-4589	151	32	.	.	PUNCT
ap-4589	152	1	(	(	PUNCT
ap-4589	152	2	4.4	4.4	NUM
ap-4589	152	3	)	)	PUNCT
ap-4589	152	4	the	the	DET
ap-4589	152	5	invariant	invariant	ADJ
ap-4589	152	6	operator	operator	NOUN
ap-4589	152	7	is	be	AUX
ap-4589	152	8	given	give	VERB
ap-4589	152	9	by	by	ADP
ap-4589	152	10	h	h	NOUN
ap-4589	152	11	=	=	SYM
ap-4589	152	12	j−j+	j−j+	PROPN
ap-4589	153	1	−	−	PROPN
ap-4589	153	2	j2	j2	PROPN
ap-4589	153	3	3	3	NUM
ap-4589	153	4	−	−	PROPN
ap-4589	153	5	g2	g2	PROPN
ap-4589	153	6	j2	j2	PROPN
ap-4589	153	7	3	3	X
ap-4589	153	8	=	=	SYM
ap-4589	153	9	j+j−	j+j−	PROPN
ap-4589	153	10	−	−	PROPN
ap-4589	153	11	(	(	PUNCT
ap-4589	153	12	j3	j3	PROPN
ap-4589	153	13	−	−	PROPN
ap-4589	153	14	1)2	1)2	PROPN
ap-4589	153	15	−	−	PROPN
ap-4589	154	1	g2	g2	PROPN
ap-4589	154	2	(	(	PUNCT
ap-4589	154	3	j3	j3	PROPN
ap-4589	154	4	−	−	PROPN
ap-4589	154	5	1)2	1)2	NUM
ap-4589	154	6	=	=	PUNCT
ap-4589	154	7	−∂2	−∂2	NOUN
ap-4589	154	8	x	x	PUNCT
ap-4589	155	1	+	+	PUNCT
ap-4589	155	2	j3(j3	j3(j3	NUM
ap-4589	155	3	−	−	PROPN
ap-4589	155	4	1	1	NUM
ap-4589	155	5	)	)	PUNCT
ap-4589	155	6	sinh2	sinh2	NOUN
ap-4589	155	7	x	x	PUNCT
ap-4589	156	1	−	−	NOUN
ap-4589	156	2	2	2	NUM
ap-4589	156	3	g	g	NOUN
ap-4589	156	4	coth	coth	NOUN
ap-4589	156	5	x.	x.	NOUN
ap-4589	156	6	(	(	PUNCT
ap-4589	156	7	4.5	4.5	NUM
ap-4589	156	8	)	)	PUNCT
ap-4589	156	9	we	we	PRON
ap-4589	156	10	note	note	VERB
ap-4589	156	11	that	that	SCONJ
ap-4589	156	12	,	,	PUNCT
ap-4589	156	13	if	if	SCONJ
ap-4589	156	14	g	g	PROPN
ap-4589	156	15	=	=	SYM
ap-4589	156	16	0	0	NUM
ap-4589	156	17	,	,	PUNCT
ap-4589	156	18	(	(	PUNCT
ap-4589	156	19	4.4	4.4	NUM
ap-4589	156	20	)	)	PUNCT
ap-4589	156	21	reduce	reduce	VERB
ap-4589	156	22	to	to	ADP
ap-4589	156	23	the	the	DET
ap-4589	156	24	standard	standard	ADJ
ap-4589	156	25	commutation	commutation	NOUN
ap-4589	156	26	relations	relation	NOUN
ap-4589	156	27	for	for	ADP
ap-4589	156	28	the	the	DET
ap-4589	156	29	lie	lie	NOUN
ap-4589	156	30	algebra	algebra	VERB
ap-4589	156	31	so(2	so(2	NOUN
ap-4589	156	32	,	,	PUNCT
ap-4589	156	33	1	1	NUM
ap-4589	156	34	)	)	PUNCT
ap-4589	156	35	under	under	ADP
ap-4589	156	36	the	the	DET
ap-4589	156	37	shift	shift	NOUN
ap-4589	156	38	j3	j3	PROPN
ap-4589	156	39	→	→	SYM
ap-4589	156	40	j3	j3	PROPN
ap-4589	156	41	+	+	PROPN
ap-4589	156	42	1/2	1/2	NUM
ap-4589	156	43	.	.	PUNCT
ap-4589	157	1	in	in	ADP
ap-4589	157	2	other	other	ADJ
ap-4589	157	3	words	word	NOUN
ap-4589	157	4	,	,	PUNCT
ap-4589	157	5	the	the	DET
ap-4589	157	6	operators	operator	NOUN
ap-4589	157	7	(	(	PUNCT
ap-4589	157	8	4.3	4.3	NUM
ap-4589	157	9	)	)	PUNCT
ap-4589	157	10	provide	provide	VERB
ap-4589	157	11	one	one	NUM
ap-4589	157	12	of	of	ADP
ap-4589	157	13	differential	differential	ADJ
ap-4589	157	14	realizations	realization	NOUN
ap-4589	157	15	of	of	ADP
ap-4589	157	16	so(2	so(2	NOUN
ap-4589	157	17	,	,	PUNCT
ap-4589	157	18	1	1	NUM
ap-4589	157	19	)	)	PUNCT
ap-4589	157	20	if	if	SCONJ
ap-4589	157	21	g	g	PROPN
ap-4589	157	22	=	=	SYM
ap-4589	157	23	0	0	PROPN
ap-4589	157	24	and	and	CCONJ
ap-4589	157	25	j3	j3	PROPN
ap-4589	157	26	→	→	SYM
ap-4589	157	27	j3	j3	PROPN
ap-4589	157	28	+	+	PROPN
ap-4589	157	29	1/2	1/2	NUM
ap-4589	157	30	.	.	PUNCT
ap-4589	158	1	unfortunately	unfortunately	ADV
ap-4589	158	2	,	,	PUNCT
ap-4589	158	3	however	however	ADV
ap-4589	158	4	,	,	PUNCT
ap-4589	158	5	this	this	DET
ap-4589	158	6	lie	lie	NOUN
ap-4589	158	7	-	-	PUNCT
ap-4589	158	8	algebraic	algebraic	ADJ
ap-4589	158	9	structure	structure	NOUN
ap-4589	158	10	is	be	AUX
ap-4589	158	11	less	less	ADV
ap-4589	158	12	useful	useful	ADJ
ap-4589	158	13	in	in	ADP
ap-4589	158	14	the	the	DET
ap-4589	158	15	present	present	ADJ
ap-4589	158	16	problem	problem	NOUN
ap-4589	158	17	because	because	SCONJ
ap-4589	158	18	the	the	DET
ap-4589	158	19	invariant	invariant	ADJ
ap-4589	158	20	operator	operator	NOUN
ap-4589	158	21	(	(	PUNCT
ap-4589	158	22	4.5	4.5	NUM
ap-4589	158	23	)	)	PUNCT
ap-4589	158	24	does	do	AUX
ap-4589	158	25	not	not	PART
ap-4589	158	26	contain	contain	VERB
ap-4589	158	27	discrete	discrete	ADJ
ap-4589	158	28	eigenvalues	eigenvalue	NOUN
ap-4589	158	29	if	if	SCONJ
ap-4589	158	30	g	g	PROPN
ap-4589	158	31	=	=	SYM
ap-4589	158	32	0	0	PROPN
ap-4589	158	33	and	and	CCONJ
ap-4589	158	34	j3	j3	PROPN
ap-4589	158	35	has	have	VERB
ap-4589	158	36	real	real	ADJ
ap-4589	158	37	eigenvalues	eigenvalue	NOUN
ap-4589	158	38	.	.	PUNCT
ap-4589	159	1	as	as	SCONJ
ap-4589	159	2	we	we	PRON
ap-4589	159	3	will	will	AUX
ap-4589	159	4	see	see	VERB
ap-4589	159	5	shortly	shortly	ADV
ap-4589	159	6	,	,	PUNCT
ap-4589	159	7	however	however	ADV
ap-4589	159	8	,	,	PUNCT
ap-4589	159	9	this	this	DET
ap-4589	159	10	situation	situation	NOUN
ap-4589	159	11	gets	get	AUX
ap-4589	159	12	changed	change	VERB
ap-4589	159	13	if	if	SCONJ
ap-4589	159	14	g	g	PROPN
ap-4589	159	15	is	be	AUX
ap-4589	159	16	non	non	ADJ
ap-4589	159	17	-	-	ADJ
ap-4589	159	18	vanishing	vanishing	ADJ
ap-4589	159	19	.	.	PUNCT
ap-4589	160	1	now	now	ADV
ap-4589	160	2	,	,	PUNCT
ap-4589	160	3	let	let	VERB
ap-4589	160	4	|e	|e	PROPN
ap-4589	160	5	,	,	PUNCT
ap-4589	160	6	j	j	PROPN
ap-4589	160	7	〉	〉	PROPN
ap-4589	160	8	be	be	AUX
ap-4589	160	9	a	a	DET
ap-4589	160	10	simultaneous	simultaneous	ADJ
ap-4589	160	11	eigenstate	eigenstate	NOUN
ap-4589	160	12	of	of	ADP
ap-4589	160	13	h	h	PROPN
ap-4589	160	14	and	and	CCONJ
ap-4589	160	15	j3	j3	PROPN
ap-4589	160	16	:	:	PUNCT
ap-4589	160	17	h|e	h|e	PROPN
ap-4589	160	18	,	,	PUNCT
ap-4589	160	19	j	j	PROPN
ap-4589	160	20	〉	〉	PROPN
ap-4589	160	21	=	=	SYM
ap-4589	160	22	e|e	e|e	PROPN
ap-4589	160	23	,	,	PUNCT
ap-4589	160	24	j	j	PROPN
ap-4589	160	25	〉	〉	PROPN
ap-4589	160	26	,	,	PUNCT
ap-4589	160	27	j3|e	j3|e	PROPN
ap-4589	160	28	,	,	PUNCT
ap-4589	160	29	j	j	PROPN
ap-4589	160	30	〉	〉	PROPN
ap-4589	160	31	=	=	SYM
ap-4589	160	32	j|e	j|e	PROPN
ap-4589	160	33	,	,	PUNCT
ap-4589	160	34	j	j	PROPN
ap-4589	160	35	〉	〉	PROPN
ap-4589	160	36	.	.	PUNCT
ap-4589	161	1	(	(	PUNCT
ap-4589	161	2	4.6	4.6	NUM
ap-4589	161	3	)	)	PUNCT
ap-4589	161	4	then	then	ADV
ap-4589	161	5	,	,	PUNCT
ap-4589	161	6	under	under	ADP
ap-4589	161	7	the	the	DET
ap-4589	161	8	normalization	normalization	NOUN
ap-4589	161	9	condition	condition	NOUN
ap-4589	161	10	‖|e	‖|e	PROPN
ap-4589	161	11	,	,	PUNCT
ap-4589	161	12	j〉‖	j〉‖	NOUN
ap-4589	161	13	=	=	SYM
ap-4589	162	1	1	1	NUM
ap-4589	162	2	,	,	PUNCT
ap-4589	162	3	the	the	DET
ap-4589	162	4	squared	squared	ADJ
ap-4589	162	5	norms	norm	NOUN
ap-4589	162	6	‖j±|e	‖j±|e	PROPN
ap-4589	162	7	,	,	PUNCT
ap-4589	162	8	j〉‖2	j〉‖2	PROPN
ap-4589	162	9	are	be	AUX
ap-4589	162	10	evaluated	evaluate	VERB
ap-4589	162	11	as	as	SCONJ
ap-4589	162	12	follows	follow	VERB
ap-4589	162	13	:	:	PUNCT
ap-4589	162	14	‖j+|e	‖j+|e	PROPN
ap-4589	162	15	,	,	PUNCT
ap-4589	162	16	j〉‖2	j〉‖2	NOUN
ap-4589	162	17	=	=	SYM
ap-4589	163	1	e	e	PROPN
ap-4589	163	2	+	+	CCONJ
ap-4589	163	3	j2	j2	PROPN
ap-4589	163	4	+	+	CCONJ
ap-4589	163	5	g2	g2	PROPN
ap-4589	163	6	j2	j2	PROPN
ap-4589	163	7	≥	≥	PROPN
ap-4589	163	8	0	0	NUM
ap-4589	163	9	,	,	PUNCT
ap-4589	163	10	‖j−|e	‖j−|e	NUM
ap-4589	163	11	,	,	PUNCT
ap-4589	163	12	j〉‖2	j〉‖2	NOUN
ap-4589	163	13	=	=	SYM
ap-4589	163	14	e	e	X
ap-4589	163	15	+	+	CCONJ
ap-4589	163	16	(	(	PUNCT
ap-4589	163	17	j	j	PROPN
ap-4589	163	18	−	−	PROPN
ap-4589	163	19	1)2	1)2	NUM
ap-4589	163	20	+	+	NUM
ap-4589	163	21	g2	g2	PROPN
ap-4589	163	22	(	(	PUNCT
ap-4589	163	23	j	j	PROPN
ap-4589	164	1	−	−	PROPN
ap-4589	164	2	1)2	1)2	NUM
ap-4589	164	3	≥	≥	NOUN
ap-4589	164	4	0	0	NUM
ap-4589	164	5	.	.	PUNCT
ap-4589	165	1	(	(	PUNCT
ap-4589	165	2	4.7	4.7	NUM
ap-4589	165	3	)	)	PUNCT
ap-4589	165	4	these	these	DET
ap-4589	165	5	conditions	condition	NOUN
ap-4589	165	6	are	be	AUX
ap-4589	165	7	enough	enough	ADJ
ap-4589	165	8	to	to	PART
ap-4589	165	9	classify	classify	VERB
ap-4589	165	10	representations	representation	NOUN
ap-4589	165	11	.	.	PUNCT
ap-4589	166	1	in	in	ADP
ap-4589	166	2	contrast	contrast	NOUN
ap-4589	166	3	to	to	ADP
ap-4589	166	4	the	the	DET
ap-4589	166	5	previous	previous	ADJ
ap-4589	166	6	two	two	NUM
ap-4589	166	7	examples	example	NOUN
ap-4589	166	8	,	,	PUNCT
ap-4589	166	9	there	there	PRON
ap-4589	166	10	are	be	VERB
ap-4589	166	11	several	several	ADJ
ap-4589	166	12	nontrivial	nontrivial	ADJ
ap-4589	166	13	representations	representation	NOUN
ap-4589	166	14	depending	depend	VERB
ap-4589	166	15	on	on	ADP
ap-4589	166	16	the	the	DET
ap-4589	166	17	range	range	NOUN
ap-4589	166	18	of	of	ADP
ap-4589	166	19	j.	j.	PROPN
ap-4589	166	20	for	for	ADP
ap-4589	166	21	g	g	PROPN
ap-4589	166	22	>	>	PROPN
ap-4589	166	23	1/4	1/4	NUM
ap-4589	166	24	,	,	PUNCT
ap-4589	166	25	we	we	PRON
ap-4589	166	26	have	have	VERB
ap-4589	166	27	the	the	DET
ap-4589	166	28	following	follow	VERB
ap-4589	166	29	three	three	NUM
ap-4589	166	30	distinct	distinct	ADJ
ap-4589	166	31	representations	representation	NOUN
ap-4589	166	32	(	(	PUNCT
ap-4589	166	33	see	see	VERB
ap-4589	166	34	figure	figure	NOUN
ap-4589	166	35	2(c	2(c	NUM
ap-4589	166	36	)	)	PUNCT
ap-4589	166	37	):	):	PUNCT
ap-4589	166	38	•	•	NUM
ap-4589	166	39	case	case	NOUN
ap-4589	166	40	j	j	PROPN
ap-4589	166	41	∈	∈	PROPN
ap-4589	166	42	(	(	PUNCT
ap-4589	166	43	−∞,−√g	−∞,−√g	NOUN
ap-4589	166	44	):	):	PUNCT
ap-4589	166	45	infinite	infinite	ADJ
ap-4589	166	46	-	-	PUNCT
ap-4589	166	47	dimensional	dimensional	ADJ
ap-4589	166	48	representation	representation	NOUN
ap-4589	166	49	.	.	PUNCT
ap-4589	167	1	let	let	VERB
ap-4589	167	2	ν	ν	X
ap-4589	167	3	∈	∈	PROPN
ap-4589	167	4	(	(	PUNCT
ap-4589	167	5	−∞,−√g	−∞,−√g	NOUN
ap-4589	167	6	)	)	PUNCT
ap-4589	167	7	be	be	AUX
ap-4589	167	8	fixed	fix	VERB
ap-4589	167	9	.	.	PUNCT
ap-4589	168	1	then	then	ADV
ap-4589	168	2	the	the	DET
ap-4589	168	3	representation	representation	NOUN
ap-4589	168	4	space	space	NOUN
ap-4589	168	5	is	be	AUX
ap-4589	168	6	spanned	span	VERB
ap-4589	168	7	by	by	ADP
ap-4589	168	8	the	the	DET
ap-4589	168	9	following	follow	VERB
ap-4589	168	10	infinitely	infinitely	ADV
ap-4589	168	11	many	many	ADJ
ap-4589	168	12	vectors	vector	NOUN
ap-4589	168	13	:	:	PUNCT
ap-4589	168	14	{	{	PUNCT
ap-4589	168	15	|e	|e	PROPN
ap-4589	168	16	,	,	PUNCT
ap-4589	168	17	j	j	PROPN
ap-4589	168	18	〉	〉	NOUN
ap-4589	168	19	:	:	PUNCT
ap-4589	168	20	e	e	X
ap-4589	168	21	=	=	PROPN
ap-4589	168	22	−ν2	−ν2	PROPN
ap-4589	168	23	−	−	PROPN
ap-4589	168	24	g2	g2	PROPN
ap-4589	168	25	ν2	ν2	NOUN
ap-4589	168	26	and	and	CCONJ
ap-4589	168	27	j	j	PROPN
ap-4589	168	28	∈	∈	PROPN
ap-4589	168	29	{	{	PUNCT
ap-4589	168	30	ν	ν	NOUN
ap-4589	168	31	,	,	PUNCT
ap-4589	168	32	ν	ν	X
ap-4589	168	33	−	−	PROPN
ap-4589	168	34	1	1	NUM
ap-4589	168	35	,	,	PUNCT
ap-4589	168	36	·	·	PUNCT
ap-4589	168	37	·	·	PUNCT
ap-4589	168	38	·	·	PUNCT
ap-4589	168	39	}	}	PUNCT
ap-4589	168	40	}	}	PUNCT
ap-4589	168	41	.	.	PUNCT
ap-4589	169	1	(	(	PUNCT
ap-4589	169	2	4.8	4.8	NUM
ap-4589	169	3	)	)	PUNCT
ap-4589	169	4	we	we	PRON
ap-4589	169	5	emphasize	emphasize	VERB
ap-4589	169	6	that	that	SCONJ
ap-4589	169	7	in	in	ADP
ap-4589	169	8	this	this	DET
ap-4589	169	9	case	case	NOUN
ap-4589	169	10	the	the	DET
ap-4589	169	11	parameter	parameter	NOUN
ap-4589	169	12	ν	ν	X
ap-4589	169	13	∈	∈	PROPN
ap-4589	169	14	(	(	PUNCT
ap-4589	169	15	−∞,−√g	−∞,−√g	NOUN
ap-4589	169	16	)	)	PUNCT
ap-4589	169	17	is	be	AUX
ap-4589	169	18	not	not	PART
ap-4589	169	19	necessarily	necessarily	ADV
ap-4589	169	20	restricted	restrict	VERB
ap-4589	169	21	to	to	ADP
ap-4589	169	22	an	an	DET
ap-4589	169	23	integer	integer	NOUN
ap-4589	169	24	or	or	CCONJ
ap-4589	169	25	half	half	NOUN
ap-4589	169	26	-	-	PUNCT
ap-4589	169	27	integer	integer	NOUN
ap-4589	169	28	.	.	PUNCT
ap-4589	170	1	this	this	PRON
ap-4589	170	2	is	be	AUX
ap-4589	170	3	a	a	DET
ap-4589	170	4	one	one	NUM
ap-4589	170	5	-	-	PUNCT
ap-4589	170	6	parameter	parameter	NOUN
ap-4589	170	7	family	family	NOUN
ap-4589	170	8	of	of	ADP
ap-4589	170	9	infinite	infinite	ADJ
ap-4589	170	10	-	-	PUNCT
ap-4589	170	11	dimensional	dimensional	ADJ
ap-4589	170	12	representation	representation	NOUN
ap-4589	170	13	of	of	ADP
ap-4589	170	14	the	the	DET
ap-4589	170	15	algebraic	algebraic	ADJ
ap-4589	170	16	system	system	NOUN
ap-4589	170	17	{	{	PUNCT
ap-4589	170	18	j3	j3	PROPN
ap-4589	170	19	,	,	PUNCT
ap-4589	170	20	j+	j+	NUM
ap-4589	170	21	,	,	PUNCT
ap-4589	170	22	j−	j−	PROPN
ap-4589	170	23	}	}	PUNCT
ap-4589	170	24	.	.	PUNCT
ap-4589	171	1	•	•	NUM
ap-4589	171	2	case	case	NOUN
ap-4589	171	3	j	j	PROPN
ap-4589	171	4	∈	∈	PROPN
ap-4589	171	5	(	(	PUNCT
ap-4589	171	6	1−√g,√g	1−√g,√g	NUM
ap-4589	171	7	):	):	PUNCT
ap-4589	171	8	finite	finite	ADJ
ap-4589	171	9	-	-	ADJ
ap-4589	171	10	dimensional	dimensional	ADJ
ap-4589	171	11	representation	representation	NOUN
ap-4589	171	12	.	.	PUNCT
ap-4589	172	1	let	let	VERB
ap-4589	172	2	ν	ν	X
ap-4589	172	3	∈	∈	PROPN
ap-4589	172	4	{	{	PUNCT
ap-4589	172	5	1	1	NUM
ap-4589	172	6	2	2	NUM
ap-4589	172	7	,	,	PUNCT
ap-4589	172	8	3	3	NUM
ap-4589	172	9	2	2	NUM
ap-4589	172	10	,	,	PUNCT
ap-4589	172	11	·	·	PUNCT
ap-4589	172	12	·	·	PUNCT
ap-4589	172	13	·	·	PUNCT
ap-4589	172	14	,	,	PUNCT
ap-4589	172	15	νmax	νmax	ADV
ap-4589	172	16	}	}	PUNCT
ap-4589	172	17	be	be	AUX
ap-4589	172	18	fixed	fix	VERB
ap-4589	172	19	,	,	PUNCT
ap-4589	172	20	where	where	SCONJ
ap-4589	172	21	νmax	νmax	ADV
ap-4589	172	22	is	be	AUX
ap-4589	172	23	the	the	DET
ap-4589	172	24	maximal	maximal	ADJ
ap-4589	172	25	half	half	ADJ
ap-4589	172	26	-	-	PUNCT
ap-4589	172	27	integer	integer	NOUN
ap-4589	172	28	smaller	small	ADJ
ap-4589	172	29	than	than	ADP
ap-4589	172	30	√g	√g	ADJ
ap-4589	172	31	;	;	PUNCT
ap-4589	172	32	i.e.	i.e.	X
ap-4589	172	33	,	,	PUNCT
ap-4589	172	34	νmax	νmax	NOUN
ap-4589	172	35	=	=	SYM
ap-4589	172	36	max{ν	max{ν	PROPN
ap-4589	172	37	∈	∈	PROPN
ap-4589	172	38	1	1	NUM
ap-4589	172	39	2n	2n	NUM
ap-4589	172	40	:	:	PUNCT
ap-4589	172	41	ν	ν	X
ap-4589	172	42	<	<	X
ap-4589	172	43	√	√	X
ap-4589	172	44	g	g	NOUN
ap-4589	172	45	}	}	PUNCT
ap-4589	172	46	.	.	PUNCT
ap-4589	173	1	then	then	ADV
ap-4589	173	2	the	the	DET
ap-4589	173	3	representation	representation	NOUN
ap-4589	173	4	space	space	NOUN
ap-4589	173	5	is	be	AUX
ap-4589	173	6	spanned	span	VERB
ap-4589	173	7	by	by	ADP
ap-4589	173	8	the	the	DET
ap-4589	173	9	following	follow	VERB
ap-4589	173	10	2ν	2ν	NUM
ap-4589	173	11	vectors	vector	NOUN
ap-4589	173	12	:	:	PUNCT
ap-4589	173	13	{	{	PUNCT
ap-4589	173	14	|e	|e	PROPN
ap-4589	173	15	,	,	PUNCT
ap-4589	173	16	j	j	PROPN
ap-4589	173	17	〉	〉	NOUN
ap-4589	173	18	:	:	PUNCT
ap-4589	173	19	e	e	X
ap-4589	173	20	=	=	PROPN
ap-4589	173	21	−ν2	−ν2	PROPN
ap-4589	173	22	−	−	PROPN
ap-4589	173	23	g2	g2	PROPN
ap-4589	173	24	ν2	ν2	NOUN
ap-4589	173	25	and	and	CCONJ
ap-4589	173	26	j	j	PROPN
ap-4589	173	27	∈	∈	PROPN
ap-4589	173	28	{	{	PUNCT
ap-4589	173	29	ν	ν	NOUN
ap-4589	173	30	,	,	PUNCT
ap-4589	173	31	ν	ν	X
ap-4589	173	32	−	−	PROPN
ap-4589	173	33	1	1	NUM
ap-4589	173	34	,	,	PUNCT
ap-4589	173	35	·	·	PUNCT
ap-4589	173	36	·	·	PUNCT
ap-4589	173	37	·	·	PUNCT
ap-4589	173	38	,	,	PUNCT
ap-4589	173	39	1−	1−	NUM
ap-4589	173	40	ν	ν	X
ap-4589	173	41	}	}	PUNCT
ap-4589	173	42	}	}	PUNCT
ap-4589	173	43	.	.	PUNCT
ap-4589	174	1	(	(	PUNCT
ap-4589	174	2	4.9	4.9	NUM
ap-4589	174	3	)	)	PUNCT
ap-4589	174	4	this	this	PRON
ap-4589	174	5	is	be	AUX
ap-4589	174	6	a	a	DET
ap-4589	174	7	2ν	2ν	NUM
ap-4589	174	8	-	-	PUNCT
ap-4589	174	9	dimensional	dimensional	ADJ
ap-4589	174	10	representation	representation	NOUN
ap-4589	174	11	of	of	ADP
ap-4589	174	12	the	the	DET
ap-4589	174	13	algebraic	algebraic	ADJ
ap-4589	174	14	system	system	NOUN
ap-4589	174	15	{	{	PUNCT
ap-4589	174	16	j3	j3	PROPN
ap-4589	174	17	,	,	PUNCT
ap-4589	174	18	j+	j+	NUM
ap-4589	174	19	,	,	PUNCT
ap-4589	174	20	j−	j−	PROPN
ap-4589	174	21	}	}	PUNCT
ap-4589	174	22	.	.	PUNCT
ap-4589	175	1	•	•	NUM
ap-4589	175	2	case	case	NOUN
ap-4589	175	3	j	j	PROPN
ap-4589	175	4	∈	∈	PROPN
ap-4589	175	5	(	(	PUNCT
ap-4589	175	6	1	1	NUM
ap-4589	175	7	+	+	NOUN
ap-4589	175	8	√g,∞	√g,∞	NOUN
ap-4589	175	9	):	):	PUNCT
ap-4589	175	10	infinite	infinite	ADJ
ap-4589	175	11	-	-	PUNCT
ap-4589	175	12	dimensional	dimensional	ADJ
ap-4589	175	13	representation	representation	NOUN
ap-4589	175	14	.	.	PUNCT
ap-4589	176	1	let	let	VERB
ap-4589	176	2	ν	ν	X
ap-4589	176	3	∈	∈	PROPN
ap-4589	176	4	(	(	PUNCT
ap-4589	176	5	1	1	NUM
ap-4589	176	6	+	+	NOUN
ap-4589	176	7	√g,∞	√g,∞	NOUN
ap-4589	176	8	)	)	PUNCT
ap-4589	176	9	be	be	AUX
ap-4589	176	10	fixed	fix	VERB
ap-4589	176	11	.	.	PUNCT
ap-4589	177	1	then	then	ADV
ap-4589	177	2	the	the	DET
ap-4589	177	3	representation	representation	NOUN
ap-4589	177	4	space	space	NOUN
ap-4589	177	5	is	be	AUX
ap-4589	177	6	spanned	span	VERB
ap-4589	177	7	by	by	ADP
ap-4589	177	8	the	the	DET
ap-4589	177	9	following	follow	VERB
ap-4589	177	10	infinitely	infinitely	ADV
ap-4589	177	11	many	many	ADJ
ap-4589	177	12	vectors	vector	NOUN
ap-4589	177	13	:	:	PUNCT
ap-4589	177	14	{	{	PUNCT
ap-4589	177	15	|e	|e	PROPN
ap-4589	177	16	,	,	PUNCT
ap-4589	177	17	j	j	PROPN
ap-4589	177	18	〉	〉	NOUN
ap-4589	177	19	:	:	PUNCT
ap-4589	177	20	e	e	X
ap-4589	177	21	=	=	PUNCT
ap-4589	177	22	−(ν	−(ν	VERB
ap-4589	177	23	−	−	PROPN
ap-4589	177	24	1)2	1)2	NUM
ap-4589	177	25	−	−	PROPN
ap-4589	177	26	g2	g2	PROPN
ap-4589	177	27	(	(	PUNCT
ap-4589	177	28	ν	ν	X
ap-4589	177	29	−	−	PROPN
ap-4589	177	30	1)2	1)2	NUM
ap-4589	177	31	and	and	CCONJ
ap-4589	177	32	j	j	PROPN
ap-4589	177	33	∈	∈	PROPN
ap-4589	177	34	{	{	PUNCT
ap-4589	177	35	ν	ν	NOUN
ap-4589	177	36	,	,	PUNCT
ap-4589	177	37	ν	ν	X
ap-4589	177	38	+	+	NOUN
ap-4589	177	39	1	1	NUM
ap-4589	177	40	,	,	PUNCT
ap-4589	177	41	·	·	PUNCT
ap-4589	177	42	·	·	PUNCT
ap-4589	177	43	·	·	PUNCT
ap-4589	177	44	}	}	PUNCT
ap-4589	177	45	}	}	PUNCT
ap-4589	177	46	.	.	PUNCT
ap-4589	178	1	(	(	PUNCT
ap-4589	178	2	4.10	4.10	NUM
ap-4589	178	3	)	)	PUNCT
ap-4589	178	4	note	note	NOUN
ap-4589	178	5	that	that	SCONJ
ap-4589	178	6	ν	ν	PROPN
ap-4589	178	7	∈	∈	PROPN
ap-4589	178	8	(	(	PUNCT
ap-4589	178	9	1	1	NUM
ap-4589	178	10	+	+	NOUN
ap-4589	178	11	√g,∞	√g,∞	NOUN
ap-4589	178	12	)	)	PUNCT
ap-4589	178	13	is	be	AUX
ap-4589	178	14	a	a	DET
ap-4589	178	15	continuous	continuous	ADJ
ap-4589	178	16	parameter	parameter	NOUN
ap-4589	178	17	and	and	CCONJ
ap-4589	178	18	is	be	AUX
ap-4589	178	19	not	not	PART
ap-4589	178	20	necessarily	necessarily	ADV
ap-4589	178	21	be	be	AUX
ap-4589	178	22	an	an	DET
ap-4589	178	23	integer	integer	NOUN
ap-4589	178	24	or	or	CCONJ
ap-4589	178	25	half	half	NOUN
ap-4589	178	26	-	-	PUNCT
ap-4589	178	27	integer	integer	NOUN
ap-4589	178	28	.	.	PUNCT
ap-4589	179	1	this	this	PRON
ap-4589	179	2	is	be	AUX
ap-4589	179	3	another	another	DET
ap-4589	179	4	one	one	NUM
ap-4589	179	5	-	-	PUNCT
ap-4589	179	6	parameter	parameter	NOUN
ap-4589	179	7	family	family	NOUN
ap-4589	179	8	of	of	ADP
ap-4589	179	9	infinite	infinite	ADJ
ap-4589	179	10	-	-	PUNCT
ap-4589	179	11	dimensional	dimensional	ADJ
ap-4589	179	12	representation	representation	NOUN
ap-4589	179	13	of	of	ADP
ap-4589	179	14	the	the	DET
ap-4589	179	15	algebraic	algebraic	ADJ
ap-4589	179	16	system	system	NOUN
ap-4589	179	17	{	{	PUNCT
ap-4589	179	18	j3	j3	PROPN
ap-4589	179	19	,	,	PUNCT
ap-4589	179	20	j+	j+	NUM
ap-4589	179	21	,	,	PUNCT
ap-4589	179	22	j−	j−	PROPN
ap-4589	179	23	}	}	PUNCT
ap-4589	179	24	.	.	PUNCT
ap-4589	180	1	451	451	NUM
ap-4589	180	2	satoshi	satoshi	PROPN
ap-4589	180	3	ohya	ohya	PROPN
ap-4589	180	4	acta	acta	PROPN
ap-4589	180	5	polytechnica	polytechnica	PROPN
ap-4589	180	6	one	one	PRON
ap-4589	180	7	may	may	AUX
ap-4589	180	8	notice	notice	VERB
ap-4589	180	9	that	that	SCONJ
ap-4589	180	10	the	the	DET
ap-4589	180	11	region	region	NOUN
ap-4589	180	12	[	[	X
ap-4589	180	13	−√g	−√g	PROPN
ap-4589	180	14	,	,	PUNCT
ap-4589	180	15	1−√g	1−√g	NUM
ap-4589	180	16	]	]	PUNCT
ap-4589	180	17	∪	∪	ADP
ap-4589	180	18	[	[	X
ap-4589	180	19	√g	√g	ADJ
ap-4589	180	20	,	,	PUNCT
ap-4589	180	21	1	1	NUM
ap-4589	180	22	+	+	NOUN
ap-4589	180	23	√g	√g	X
ap-4589	180	24	]	]	PUNCT
ap-4589	180	25	is	be	AUX
ap-4589	180	26	excluded	exclude	VERB
ap-4589	180	27	in	in	ADP
ap-4589	180	28	the	the	DET
ap-4589	180	29	above	above	ADJ
ap-4589	180	30	classification	classification	NOUN
ap-4589	180	31	.	.	PUNCT
ap-4589	181	1	this	this	PRON
ap-4589	181	2	is	be	AUX
ap-4589	181	3	because	because	SCONJ
ap-4589	181	4	there	there	PRON
ap-4589	181	5	is	be	VERB
ap-4589	181	6	no	no	DET
ap-4589	181	7	bound	bound	ADJ
ap-4589	181	8	state	state	NOUN
ap-4589	181	9	in	in	ADP
ap-4589	181	10	this	this	DET
ap-4589	181	11	region	region	NOUN
ap-4589	181	12	for	for	ADP
ap-4589	181	13	both	both	CCONJ
ap-4589	181	14	the	the	DET
ap-4589	181	15	hyperbolic	hyperbolic	ADJ
ap-4589	181	16	kepler	kepler	NOUN
ap-4589	181	17	and	and	CCONJ
ap-4589	181	18	rosen	rosen	PROPN
ap-4589	181	19	–	–	PUNCT
ap-4589	181	20	morse	morse	ADJ
ap-4589	181	21	potential	potential	ADJ
ap-4589	181	22	problems	problem	NOUN
ap-4589	181	23	.	.	PUNCT
ap-4589	182	1	we	we	PRON
ap-4589	182	2	note	note	VERB
ap-4589	182	3	that	that	SCONJ
ap-4589	182	4	the	the	DET
ap-4589	182	5	finite	finite	ADJ
ap-4589	182	6	-	-	ADJ
ap-4589	182	7	dimensional	dimensional	ADJ
ap-4589	182	8	representation	representation	NOUN
ap-4589	182	9	(	(	PUNCT
ap-4589	182	10	4.9	4.9	NUM
ap-4589	182	11	)	)	PUNCT
ap-4589	182	12	disappears	disappear	VERB
ap-4589	182	13	for	for	ADP
ap-4589	182	14	g	g	PROPN
ap-4589	182	15	≤	≤	NUM
ap-4589	182	16	1/4	1/4	NUM
ap-4589	182	17	,	,	PUNCT
ap-4589	182	18	whereas	whereas	SCONJ
ap-4589	182	19	the	the	DET
ap-4589	182	20	infinite	infinite	ADJ
ap-4589	182	21	-	-	PUNCT
ap-4589	182	22	dimensional	dimensional	ADJ
ap-4589	182	23	representations	representation	NOUN
ap-4589	182	24	(	(	PUNCT
ap-4589	182	25	4.8	4.8	NUM
ap-4589	182	26	)	)	PUNCT
ap-4589	182	27	and	and	CCONJ
ap-4589	182	28	(	(	PUNCT
ap-4589	182	29	4.10	4.10	NUM
ap-4589	182	30	)	)	PUNCT
ap-4589	182	31	remain	remain	VERB
ap-4589	182	32	present	present	ADJ
ap-4589	182	33	for	for	ADP
ap-4589	182	34	g	g	PROPN
ap-4589	182	35	≤	≤	NUM
ap-4589	182	36	1/4	1/4	NUM
ap-4589	182	37	.	.	PUNCT
ap-4589	183	1	now	now	ADV
ap-4589	183	2	we	we	PRON
ap-4589	183	3	have	have	AUX
ap-4589	183	4	classified	classify	VERB
ap-4589	183	5	the	the	DET
ap-4589	183	6	representations	representation	NOUN
ap-4589	183	7	of	of	ADP
ap-4589	183	8	the	the	DET
ap-4589	183	9	potential	potential	ADJ
ap-4589	183	10	algebra	algebra	NOUN
ap-4589	183	11	.	.	PUNCT
ap-4589	184	1	the	the	DET
ap-4589	184	2	next	next	ADJ
ap-4589	184	3	task	task	NOUN
ap-4589	184	4	we	we	PRON
ap-4589	184	5	have	have	VERB
ap-4589	184	6	to	to	PART
ap-4589	184	7	do	do	VERB
ap-4589	184	8	is	be	AUX
ap-4589	184	9	to	to	PART
ap-4589	184	10	understand	understand	VERB
ap-4589	184	11	which	which	DET
ap-4589	184	12	representations	representation	NOUN
ap-4589	184	13	are	be	AUX
ap-4589	184	14	realized	realize	VERB
ap-4589	184	15	in	in	ADP
ap-4589	184	16	the	the	DET
ap-4589	184	17	hyperbolic	hyperbolic	ADJ
ap-4589	184	18	kepler	kepler	PROPN
ap-4589	184	19	problem	problem	NOUN
ap-4589	184	20	.	.	PUNCT
ap-4589	185	1	to	to	PART
ap-4589	185	2	see	see	VERB
ap-4589	185	3	this	this	PRON
ap-4589	185	4	,	,	PUNCT
ap-4589	185	5	let	let	VERB
ap-4589	185	6	us	we	PRON
ap-4589	185	7	consider	consider	VERB
ap-4589	185	8	the	the	DET
ap-4589	185	9	potential	potential	ADJ
ap-4589	185	10	v	v	X
ap-4589	185	11	(	(	PUNCT
ap-4589	185	12	x	x	NOUN
ap-4589	185	13	)	)	PUNCT
ap-4589	185	14	=	=	SYM
ap-4589	186	1	j(j	j(j	PROPN
ap-4589	186	2	−	−	PROPN
ap-4589	187	1	1)/	1)/	PROPN
ap-4589	187	2	sinh2	sinh2	NOUN
ap-4589	187	3	x−	x−	PROPN
ap-4589	187	4	2	2	NUM
ap-4589	187	5	g	g	NOUN
ap-4589	187	6	coth	coth	NOUN
ap-4589	187	7	x.	x.	NOUN
ap-4589	187	8	in	in	ADP
ap-4589	187	9	order	order	NOUN
ap-4589	187	10	to	to	PART
ap-4589	187	11	have	have	AUX
ap-4589	187	12	a	a	DET
ap-4589	187	13	bound	bind	VERB
ap-4589	187	14	state	state	NOUN
ap-4589	187	15	,	,	PUNCT
ap-4589	187	16	it	it	PRON
ap-4589	187	17	is	be	AUX
ap-4589	187	18	necessary	necessary	ADJ
ap-4589	187	19	that	that	SCONJ
ap-4589	187	20	v	v	X
ap-4589	187	21	(	(	PUNCT
ap-4589	187	22	x	x	X
ap-4589	187	23	)	)	PUNCT
ap-4589	187	24	has	have	VERB
ap-4589	187	25	a	a	DET
ap-4589	187	26	minimum	minimum	NOUN
ap-4589	187	27	on	on	ADP
ap-4589	187	28	the	the	DET
ap-4589	187	29	half	half	NOUN
ap-4589	187	30	line.5	line.5	VERB
ap-4589	187	31	this	this	PRON
ap-4589	187	32	is	be	AUX
ap-4589	187	33	achieved	achieve	VERB
ap-4589	187	34	if	if	SCONJ
ap-4589	187	35	and	and	CCONJ
ap-4589	187	36	only	only	ADV
ap-4589	187	37	if	if	SCONJ
ap-4589	187	38	j	j	PROPN
ap-4589	187	39	is	be	AUX
ap-4589	187	40	in	in	ADP
ap-4589	187	41	the	the	DET
ap-4589	187	42	range	range	NOUN
ap-4589	187	43	(	(	PUNCT
ap-4589	187	44	1	1	NUM
ap-4589	187	45	2	2	NUM
ap-4589	187	46	−	−	NOUN
ap-4589	187	47	√	√	NUM
ap-4589	188	1	g	g	NOUN
ap-4589	188	2	+	+	CCONJ
ap-4589	188	3	1	1	NUM
ap-4589	188	4	4	4	NUM
ap-4589	188	5	,	,	PUNCT
ap-4589	188	6	1	1	NUM
ap-4589	188	7	2	2	NUM
ap-4589	188	8	+	+	CCONJ
ap-4589	188	9	√	√	PROPN
ap-4589	188	10	g	g	NOUN
ap-4589	188	11	+	+	CCONJ
ap-4589	188	12	1	1	NUM
ap-4589	188	13	4	4	NUM
ap-4589	188	14	)	)	PUNCT
ap-4589	188	15	,	,	PUNCT
ap-4589	188	16	which	which	PRON
ap-4589	188	17	includes	include	VERB
ap-4589	188	18	(	(	PUNCT
ap-4589	188	19	1−√g,√g	1−√g,√g	NUM
ap-4589	188	20	)	)	PUNCT
ap-4589	188	21	;	;	PUNCT
ap-4589	188	22	see	see	VERB
ap-4589	188	23	figure	figure	NOUN
ap-4589	188	24	2(c	2(c	NUM
ap-4589	188	25	)	)	PUNCT
ap-4589	188	26	.	.	PUNCT
ap-4589	189	1	hence	hence	ADV
ap-4589	189	2	the	the	DET
ap-4589	189	3	bound	bound	ADJ
ap-4589	189	4	state	state	NOUN
ap-4589	189	5	spectrum	spectrum	NOUN
ap-4589	189	6	should	should	AUX
ap-4589	189	7	be	be	AUX
ap-4589	189	8	related	relate	VERB
ap-4589	189	9	to	to	ADP
ap-4589	189	10	the	the	DET
ap-4589	189	11	finite	finite	ADJ
ap-4589	189	12	-	-	ADJ
ap-4589	189	13	dimensional	dimensional	ADJ
ap-4589	189	14	representation	representation	NOUN
ap-4589	189	15	(	(	PUNCT
ap-4589	189	16	4.9	4.9	NUM
ap-4589	189	17	)	)	PUNCT
ap-4589	189	18	.	.	PUNCT
ap-4589	190	1	now	now	ADV
ap-4589	190	2	it	it	PRON
ap-4589	190	3	is	be	AUX
ap-4589	190	4	easy	easy	ADJ
ap-4589	190	5	to	to	PART
ap-4589	190	6	solve	solve	VERB
ap-4589	190	7	the	the	DET
ap-4589	190	8	original	original	ADJ
ap-4589	190	9	eigenvalue	eigenvalue	NOUN
ap-4589	190	10	problem	problem	NOUN
ap-4589	190	11	hhyperbolic	hhyperbolic	ADJ
ap-4589	190	12	keplerψen	keplerψen	NOUN
ap-4589	190	13	,	,	PUNCT
ap-4589	190	14	j	j	PROPN
ap-4589	190	15	=	=	PUNCT
ap-4589	190	16	enψen	enψen	PROPN
ap-4589	190	17	,	,	PUNCT
ap-4589	190	18	j	j	PROPN
ap-4589	190	19	for	for	ADP
ap-4589	190	20	the	the	DET
ap-4589	190	21	hyperbolic	hyperbolic	ADJ
ap-4589	190	22	kepler	kepler	NOUN
ap-4589	190	23	hamiltonian	hamiltonian	NOUN
ap-4589	190	24	.	.	PUNCT
ap-4589	191	1	for	for	ADP
ap-4589	191	2	fixed	fix	VERB
ap-4589	191	3	j	j	PROPN
ap-4589	191	4	∈	∈	PROPN
ap-4589	191	5	{	{	PUNCT
ap-4589	191	6	1	1	NUM
ap-4589	191	7	2	2	NUM
ap-4589	191	8	,	,	PUNCT
ap-4589	191	9	3	3	NUM
ap-4589	191	10	2	2	NUM
ap-4589	191	11	,	,	PUNCT
ap-4589	191	12	·	·	PUNCT
ap-4589	191	13	·	·	PUNCT
ap-4589	191	14	·	·	PUNCT
ap-4589	191	15	,	,	PUNCT
ap-4589	191	16	νmax	νmax	ADV
ap-4589	191	17	}	}	PUNCT
ap-4589	191	18	,	,	PUNCT
ap-4589	191	19	the	the	DET
ap-4589	191	20	energy	energy	NOUN
ap-4589	191	21	eigenvalues	eigenvalue	NOUN
ap-4589	191	22	and	and	CCONJ
ap-4589	191	23	eigenfunctions	eigenfunction	NOUN
ap-4589	191	24	are	be	AUX
ap-4589	191	25	given	give	VERB
ap-4589	191	26	by	by	ADP
ap-4589	191	27	en	en	X
ap-4589	191	28	=	=	SYM
ap-4589	191	29	−(j	−(j	PROPN
ap-4589	191	30	+	+	CCONJ
ap-4589	191	31	n)2	n)2	PROPN
ap-4589	191	32	−	−	PROPN
ap-4589	191	33	g2	g2	PROPN
ap-4589	191	34	(	(	PUNCT
ap-4589	191	35	j	j	PROPN
ap-4589	191	36	+	+	PROPN
ap-4589	191	37	n)2	n)2	PROPN
ap-4589	191	38	,	,	PUNCT
ap-4589	191	39	n	n	CCONJ
ap-4589	191	40	∈	∈	PROPN
ap-4589	191	41	{	{	PUNCT
ap-4589	191	42	0	0	NUM
ap-4589	191	43	,	,	PUNCT
ap-4589	191	44	1	1	NUM
ap-4589	191	45	,	,	PUNCT
ap-4589	191	46	·	·	PUNCT
ap-4589	191	47	·	·	PUNCT
ap-4589	191	48	·	·	PUNCT
ap-4589	191	49	,	,	PUNCT
ap-4589	191	50	n	n	CCONJ
ap-4589	191	51	}	}	PUNCT
ap-4589	191	52	,	,	PUNCT
ap-4589	191	53	(	(	PUNCT
ap-4589	191	54	4.11	4.11	NUM
ap-4589	191	55	)	)	PUNCT
ap-4589	191	56	and	and	CCONJ
ap-4589	191	57	ψen	ψen	NOUN
ap-4589	191	58	,	,	PUNCT
ap-4589	191	59	j(x	j(x	PROPN
ap-4589	191	60	)	)	PUNCT
ap-4589	191	61	∝	∝	X
ap-4589	192	1	a−(j)a−(j	a−(j)a−(j	PRON
ap-4589	192	2	+	+	NUM
ap-4589	192	3	1	1	NUM
ap-4589	192	4	)	)	PUNCT
ap-4589	192	5	·	·	PUNCT
ap-4589	192	6	·	·	PUNCT
ap-4589	193	1	·	·	PUNCT
ap-4589	193	2	a−(j	a−(j	PROPN
ap-4589	193	3	+	+	CCONJ
ap-4589	193	4	n−	n−	NOUN
ap-4589	193	5	1)ψen	1)ψen	NUM
ap-4589	193	6	,	,	PUNCT
ap-4589	193	7	j+n(x	j+n(x	NOUN
ap-4589	193	8	)	)	PUNCT
ap-4589	193	9	,	,	PUNCT
ap-4589	193	10	(	(	PUNCT
ap-4589	193	11	4.12	4.12	NUM
ap-4589	193	12	)	)	PUNCT
ap-4589	193	13	where	where	SCONJ
ap-4589	193	14	n	n	NOUN
ap-4589	193	15	=	=	SYM
ap-4589	193	16	max{n	max{n	CCONJ
ap-4589	193	17	∈	∈	PROPN
ap-4589	193	18	z≥0	z≥0	NOUN
ap-4589	193	19	:	:	PUNCT
ap-4589	193	20	j	j	PROPN
ap-4589	193	21	+	+	CCONJ
ap-4589	193	22	n	n	CCONJ
ap-4589	193	23	<	<	X
ap-4589	193	24	√	√	PROPN
ap-4589	193	25	g	g	NOUN
ap-4589	193	26	}	}	PUNCT
ap-4589	193	27	=	=	PUNCT
ap-4589	193	28	νmax	νmax	ADV
ap-4589	193	29	−	−	PROPN
ap-4589	193	30	j	j	PROPN
ap-4589	193	31	and	and	CCONJ
ap-4589	193	32	ψen	ψen	NOUN
ap-4589	193	33	,	,	PUNCT
ap-4589	193	34	j+n(x	j+n(x	NOUN
ap-4589	193	35	)	)	PUNCT
ap-4589	193	36	∝	∝	PROPN
ap-4589	193	37	(	(	PUNCT
ap-4589	193	38	sinh	sinh	PROPN
ap-4589	193	39	x)j+n	x)j+n	PROPN
ap-4589	193	40	exp(−	exp(−	PROPN
ap-4589	193	41	g	g	PROPN
ap-4589	193	42	j+nx	j+nx	VERB
ap-4589	193	43	)	)	PUNCT
ap-4589	193	44	.	.	PUNCT
ap-4589	194	1	notice	notice	VERB
ap-4589	194	2	that	that	SCONJ
ap-4589	194	3	these	these	DET
ap-4589	194	4	results	result	NOUN
ap-4589	194	5	are	be	AUX
ap-4589	194	6	consistent	consistent	ADJ
ap-4589	194	7	with	with	ADP
ap-4589	194	8	the	the	DET
ap-4589	194	9	known	know	VERB
ap-4589	194	10	results	result	NOUN
ap-4589	194	11	[	[	X
ap-4589	194	12	5	5	NUM
ap-4589	194	13	]	]	PUNCT
ap-4589	194	14	.	.	PUNCT
ap-4589	195	1	before	before	ADP
ap-4589	195	2	closing	close	VERB
ap-4589	195	3	this	this	DET
ap-4589	195	4	subsection	subsection	NOUN
ap-4589	195	5	it	it	PRON
ap-4589	195	6	is	be	AUX
ap-4589	195	7	worthwhile	worthwhile	ADJ
ap-4589	195	8	to	to	PART
ap-4589	195	9	comment	comment	VERB
ap-4589	195	10	on	on	ADP
ap-4589	195	11	the	the	DET
ap-4589	195	12	case	case	NOUN
ap-4589	195	13	g	g	NOUN
ap-4589	195	14	≤	≤	NOUN
ap-4589	195	15	1/4	1/4	NUM
ap-4589	195	16	.	.	PUNCT
ap-4589	196	1	as	as	SCONJ
ap-4589	196	2	mentioned	mention	VERB
ap-4589	196	3	before	before	ADV
ap-4589	196	4	,	,	PUNCT
ap-4589	196	5	the	the	DET
ap-4589	196	6	finite	finite	ADJ
ap-4589	196	7	-	-	ADJ
ap-4589	196	8	dimensional	dimensional	ADJ
ap-4589	196	9	representation	representation	NOUN
ap-4589	196	10	(	(	PUNCT
ap-4589	196	11	4.9	4.9	NUM
ap-4589	196	12	)	)	PUNCT
ap-4589	196	13	disappears	disappear	VERB
ap-4589	196	14	for	for	ADP
ap-4589	196	15	g	g	PROPN
ap-4589	196	16	≤	≤	NUM
ap-4589	196	17	1/4	1/4	NUM
ap-4589	196	18	.	.	PUNCT
ap-4589	197	1	however	however	ADV
ap-4589	197	2	,	,	PUNCT
ap-4589	197	3	new	new	ADJ
ap-4589	197	4	finite	finite	ADJ
ap-4589	197	5	-	-	ADJ
ap-4589	197	6	dimensional	dimensional	ADJ
ap-4589	197	7	representations	representation	NOUN
ap-4589	197	8	appear	appear	VERB
ap-4589	197	9	in	in	ADP
ap-4589	197	10	this	this	DET
ap-4589	197	11	case	case	NOUN
ap-4589	197	12	.	.	PUNCT
ap-4589	198	1	the	the	DET
ap-4589	198	2	relevant	relevant	ADJ
ap-4589	198	3	one	one	NOUN
ap-4589	198	4	is	be	AUX
ap-4589	198	5	the	the	DET
ap-4589	198	6	following	follow	VERB
ap-4589	198	7	one	one	NUM
ap-4589	198	8	-	-	PUNCT
ap-4589	198	9	dimensional	dimensional	ADJ
ap-4589	198	10	representation	representation	NOUN
ap-4589	198	11	spanned	span	VERB
ap-4589	198	12	by	by	ADP
ap-4589	198	13	a	a	DET
ap-4589	198	14	single	single	ADJ
ap-4589	198	15	vector	vector	NOUN
ap-4589	198	16	:	:	PUNCT
ap-4589	198	17	{	{	PUNCT
ap-4589	198	18	|e	|e	PROPN
ap-4589	198	19	,	,	PUNCT
ap-4589	198	20	j	j	PROPN
ap-4589	198	21	〉	〉	NOUN
ap-4589	198	22	:	:	PUNCT
ap-4589	199	1	e	e	X
ap-4589	199	2	=	=	SYM
ap-4589	200	1	−j2	−j2	NOUN
ap-4589	200	2	−	−	PROPN
ap-4589	200	3	g2	g2	PROPN
ap-4589	200	4	j2	j2	PROPN
ap-4589	200	5	and	and	CCONJ
ap-4589	200	6	j	j	PROPN
ap-4589	200	7	=	=	NOUN
ap-4589	200	8	1	1	NUM
ap-4589	200	9	2	2	NUM
ap-4589	200	10	−	−	NOUN
ap-4589	200	11	√	√	NOUN
ap-4589	200	12	1	1	NUM
ap-4589	200	13	4	4	NUM
ap-4589	200	14	−	−	NOUN
ap-4589	200	15	g	g	PROPN
ap-4589	200	16	}	}	PUNCT
ap-4589	200	17	,	,	PUNCT
ap-4589	200	18	(	(	PUNCT
ap-4589	200	19	4.13	4.13	NUM
ap-4589	200	20	)	)	PUNCT
ap-4589	200	21	where	where	SCONJ
ap-4589	200	22	g	g	PROPN
ap-4589	200	23	∈	∈	PROPN
ap-4589	200	24	(	(	PUNCT
ap-4589	200	25	0	0	NUM
ap-4589	200	26	,	,	PUNCT
ap-4589	200	27	1/4	1/4	NUM
ap-4589	200	28	)	)	PUNCT
ap-4589	200	29	.	.	PUNCT
ap-4589	201	1	notice	notice	VERB
ap-4589	201	2	that	that	SCONJ
ap-4589	201	3	this	this	DET
ap-4589	201	4	j	j	PROPN
ap-4589	201	5	is	be	AUX
ap-4589	201	6	one	one	NUM
ap-4589	201	7	of	of	ADP
ap-4589	201	8	the	the	DET
ap-4589	201	9	solutions	solution	NOUN
ap-4589	201	10	to	to	ADP
ap-4589	201	11	the	the	DET
ap-4589	201	12	condition	condition	NOUN
ap-4589	201	13	−j2−g2	−j2−g2	NOUN
ap-4589	201	14	/	/	SYM
ap-4589	201	15	j2	j2	PROPN
ap-4589	201	16	=	=	SYM
ap-4589	201	17	−(j−1)2−g2/(j−1)2	−(j−1)2−g2/(j−1)2	PROPN
ap-4589	201	18	.	.	PUNCT
ap-4589	202	1	now	now	ADV
ap-4589	202	2	one	one	PRON
ap-4589	202	3	can	can	AUX
ap-4589	202	4	easily	easily	ADV
ap-4589	202	5	check	check	VERB
ap-4589	202	6	that	that	SCONJ
ap-4589	202	7	this	this	DET
ap-4589	202	8	state	state	NOUN
ap-4589	202	9	vector	vector	NOUN
ap-4589	202	10	satisfies	satisfie	NOUN
ap-4589	202	11	j±|e	j±|e	PROPN
ap-4589	202	12	,	,	PUNCT
ap-4589	202	13	j	j	PROPN
ap-4589	202	14	〉	〉	NUM
ap-4589	202	15	=	=	SYM
ap-4589	202	16	0	0	X
ap-4589	202	17	.	.	PUNCT
ap-4589	203	1	it	it	PRON
ap-4589	203	2	is	be	AUX
ap-4589	203	3	also	also	ADV
ap-4589	203	4	easy	easy	ADJ
ap-4589	203	5	to	to	PART
ap-4589	203	6	see	see	VERB
ap-4589	203	7	that	that	PRON
ap-4589	203	8	,	,	PUNCT
ap-4589	203	9	for	for	ADP
ap-4589	203	10	g	g	PROPN
ap-4589	203	11	∈	∈	PROPN
ap-4589	203	12	(	(	PUNCT
ap-4589	203	13	0	0	NUM
ap-4589	203	14	,	,	PUNCT
ap-4589	203	15	1/4	1/4	NUM
ap-4589	203	16	)	)	PUNCT
ap-4589	203	17	,	,	PUNCT
ap-4589	203	18	j	j	X
ap-4589	203	19	=	=	SYM
ap-4589	203	20	1/2	1/2	NUM
ap-4589	203	21	−	−	NOUN
ap-4589	203	22	√	√	NUM
ap-4589	203	23	1/4−	1/4−	NOUN
ap-4589	203	24	g	g	NOUN
ap-4589	203	25	satisfies	satisfy	VERB
ap-4589	203	26	the	the	DET
ap-4589	203	27	condition	condition	NOUN
ap-4589	203	28	j	j	NOUN
ap-4589	203	29	<	<	X
ap-4589	203	30	√	√	PROPN
ap-4589	203	31	g	g	NOUN
ap-4589	203	32	,	,	PUNCT
ap-4589	203	33	which	which	PRON
ap-4589	203	34	is	be	AUX
ap-4589	203	35	the	the	DET
ap-4589	203	36	necessary	necessary	ADJ
ap-4589	203	37	condition	condition	NOUN
ap-4589	203	38	for	for	ADP
ap-4589	203	39	the	the	DET
ap-4589	203	40	ground	ground	NOUN
ap-4589	203	41	-	-	PUNCT
ap-4589	203	42	state	state	NOUN
ap-4589	203	43	wavefunction	wavefunction	NOUN
ap-4589	203	44	to	to	PART
ap-4589	203	45	be	be	AUX
ap-4589	203	46	normalizable	normalizable	ADJ
ap-4589	203	47	.	.	PUNCT
ap-4589	204	1	the	the	DET
ap-4589	204	2	point	point	NOUN
ap-4589	204	3	is	be	AUX
ap-4589	204	4	that	that	SCONJ
ap-4589	204	5	,	,	PUNCT
ap-4589	204	6	just	just	ADV
ap-4589	204	7	as	as	SCONJ
ap-4589	204	8	in	in	ADP
ap-4589	204	9	the	the	DET
ap-4589	204	10	case	case	NOUN
ap-4589	204	11	g	g	ADP
ap-4589	204	12	>	>	PROPN
ap-4589	204	13	1/4	1/4	NUM
ap-4589	204	14	,	,	PUNCT
ap-4589	204	15	j	j	PROPN
ap-4589	204	16	must	must	AUX
ap-4589	204	17	be	be	AUX
ap-4589	204	18	quantized	quantize	VERB
ap-4589	204	19	in	in	ADP
ap-4589	204	20	a	a	DET
ap-4589	204	21	particular	particular	ADJ
ap-4589	204	22	manner	manner	NOUN
ap-4589	204	23	in	in	ADP
ap-4589	204	24	this	this	DET
ap-4589	204	25	representation	representation	NOUN
ap-4589	204	26	theoretic	theoretic	NOUN
ap-4589	204	27	approach	approach	NOUN
ap-4589	204	28	.	.	PUNCT
ap-4589	205	1	4.2	4.2	NUM
ap-4589	205	2	.	.	PUNCT
ap-4589	206	1	rosen	rosen	PROPN
ap-4589	206	2	–	–	PUNCT
ap-4589	206	3	morse	morse	PROPN
ap-4589	206	4	let	let	VERB
ap-4589	206	5	us	we	PRON
ap-4589	206	6	finally	finally	ADV
ap-4589	206	7	move	move	VERB
ap-4589	206	8	on	on	ADP
ap-4589	206	9	to	to	ADP
ap-4589	206	10	the	the	DET
ap-4589	206	11	bound	bind	VERB
ap-4589	206	12	-	-	PUNCT
ap-4589	206	13	state	state	NOUN
ap-4589	206	14	problem	problem	NOUN
ap-4589	206	15	of	of	ADP
ap-4589	206	16	the	the	DET
ap-4589	206	17	rosen	rosen	PROPN
ap-4589	206	18	–	–	PUNCT
ap-4589	206	19	morse	morse	ADJ
ap-4589	206	20	hamiltonian	hamiltonian	NOUN
ap-4589	207	1	[	[	X
ap-4589	207	2	6	6	NUM
ap-4589	207	3	,	,	PUNCT
ap-4589	207	4	7	7	NUM
ap-4589	207	5	]	]	PUNCT
ap-4589	207	6	.	.	PUNCT
ap-4589	208	1	first	first	ADV
ap-4589	208	2	,	,	PUNCT
ap-4589	208	3	the	the	DET
ap-4589	208	4	hamiltonian	hamiltonian	ADJ
ap-4589	208	5	hrosen	hrosen	PROPN
ap-4589	208	6	–	–	PUNCT
ap-4589	208	7	morse	morse	NOUN
ap-4589	208	8	in	in	ADP
ap-4589	208	9	(	(	PUNCT
ap-4589	208	10	1.1	1.1	NUM
ap-4589	208	11	)	)	PUNCT
ap-4589	208	12	is	be	AUX
ap-4589	208	13	factorized	factorize	VERB
ap-4589	208	14	as	as	SCONJ
ap-4589	208	15	follows	follow	VERB
ap-4589	208	16	:	:	PUNCT
ap-4589	209	1	hrosen	hrosen	NOUN
ap-4589	209	2	–	–	PUNCT
ap-4589	209	3	morse	morse	ADJ
ap-4589	209	4	=	=	SYM
ap-4589	209	5	a−(j)a+(j)−	a−(j)a+(j)−	NOUN
ap-4589	209	6	j2	j2	PROPN
ap-4589	209	7	−	−	PROPN
ap-4589	209	8	g2	g2	PROPN
ap-4589	209	9	j2	j2	PROPN
ap-4589	209	10	,	,	PUNCT
ap-4589	209	11	(	(	PUNCT
ap-4589	209	12	4.14	4.14	NUM
ap-4589	209	13	)	)	PUNCT
ap-4589	210	1	where	where	SCONJ
ap-4589	210	2	a±(j	a±(j	PROPN
ap-4589	210	3	)	)	PUNCT
ap-4589	210	4	=	=	SYM
ap-4589	210	5	±	±	NUM
ap-4589	210	6	d	d	NOUN
ap-4589	210	7	dx	dx	PROPN
ap-4589	210	8	−	−	PROPN
ap-4589	210	9	j	j	PROPN
ap-4589	210	10	tanh	tanh	PROPN
ap-4589	210	11	x+	x+	PROPN
ap-4589	210	12	g	g	PROPN
ap-4589	210	13	j	j	PROPN
ap-4589	210	14	.	.	PUNCT
ap-4589	211	1	(	(	PUNCT
ap-4589	211	2	4.15	4.15	NUM
ap-4589	211	3	)	)	PUNCT
ap-4589	211	4	let	let	VERB
ap-4589	211	5	us	we	PRON
ap-4589	211	6	then	then	ADV
ap-4589	211	7	introduce	introduce	VERB
ap-4589	211	8	the	the	DET
ap-4589	211	9	following	follow	VERB
ap-4589	211	10	operators	operator	NOUN
ap-4589	211	11	:	:	PUNCT
ap-4589	211	12	j3	j3	PROPN
ap-4589	211	13	=	=	PUNCT
ap-4589	211	14	−i∂θ	−i∂θ	PROPN
ap-4589	211	15	,	,	PUNCT
ap-4589	211	16	j+	j+	NUM
ap-4589	211	17	=	=	SYM
ap-4589	212	1	eiθ	eiθ	PROPN
ap-4589	212	2	(	(	PUNCT
ap-4589	212	3	∂x	∂x	PROPN
ap-4589	212	4	−	−	PROPN
ap-4589	212	5	tanh	tanh	PROPN
ap-4589	212	6	xj3	xj3	PROPN
ap-4589	213	1	+	+	CCONJ
ap-4589	213	2	g	g	PROPN
ap-4589	213	3	j3	j3	PROPN
ap-4589	213	4	)	)	PUNCT
ap-4589	213	5	,	,	PUNCT
ap-4589	213	6	j−	j−	PROPN
ap-4589	213	7	=	=	PUNCT
ap-4589	213	8	(	(	PUNCT
ap-4589	213	9	−∂x	−∂x	NUM
ap-4589	213	10	−	−	PROPN
ap-4589	213	11	tanh	tanh	PROPN
ap-4589	213	12	xj3	xj3	PROPN
ap-4589	214	1	+	+	CCONJ
ap-4589	214	2	g	g	PROPN
ap-4589	214	3	j3	j3	PROPN
ap-4589	214	4	)	)	PUNCT
ap-4589	214	5	e−iθ	e−iθ	PROPN
ap-4589	214	6	,	,	PUNCT
ap-4589	214	7	(	(	PUNCT
ap-4589	214	8	4.16	4.16	NUM
ap-4589	214	9	)	)	PUNCT
ap-4589	214	10	which	which	PRON
ap-4589	214	11	satisfy	satisfy	VERB
ap-4589	214	12	the	the	DET
ap-4589	214	13	commutation	commutation	NOUN
ap-4589	214	14	relations	relation	NOUN
ap-4589	214	15	:	:	PUNCT
ap-4589	215	1	[	[	X
ap-4589	215	2	j3	j3	PROPN
ap-4589	215	3	,	,	PUNCT
ap-4589	215	4	j±	j±	PROPN
ap-4589	215	5	]	]	X
ap-4589	215	6	=	=	SYM
ap-4589	215	7	±j±	±j±	PROPN
ap-4589	215	8	,	,	PUNCT
ap-4589	215	9	[	[	X
ap-4589	215	10	j+	j+	NUM
ap-4589	215	11	,	,	PUNCT
ap-4589	215	12	j−	j−	PROPN
ap-4589	215	13	]	]	X
ap-4589	215	14	=	=	SYM
ap-4589	215	15	−j2	−j2	NOUN
ap-4589	215	16	3	3	NUM
ap-4589	215	17	−	−	PROPN
ap-4589	215	18	g2	g2	PROPN
ap-4589	215	19	j2	j2	PROPN
ap-4589	215	20	3	3	NUM
ap-4589	215	21	+	+	CCONJ
ap-4589	215	22	(	(	PUNCT
ap-4589	215	23	j3	j3	PROPN
ap-4589	215	24	−	−	PROPN
ap-4589	215	25	1)2	1)2	PROPN
ap-4589	215	26	+	+	CCONJ
ap-4589	215	27	g2	g2	PROPN
ap-4589	215	28	(	(	PUNCT
ap-4589	215	29	j3	j3	PROPN
ap-4589	215	30	−	−	PROPN
ap-4589	215	31	1)2	1)2	PROPN
ap-4589	215	32	.	.	PUNCT
ap-4589	216	1	(	(	PUNCT
ap-4589	216	2	4.17	4.17	NUM
ap-4589	216	3	)	)	PUNCT
ap-4589	216	4	the	the	DET
ap-4589	216	5	invariant	invariant	ADJ
ap-4589	216	6	operator	operator	NOUN
ap-4589	216	7	is	be	AUX
ap-4589	216	8	h	h	NOUN
ap-4589	216	9	=	=	SYM
ap-4589	217	1	j−j+	j−j+	PROPN
ap-4589	217	2	−	−	PROPN
ap-4589	217	3	j2	j2	PROPN
ap-4589	217	4	3	3	NUM
ap-4589	217	5	−	−	PROPN
ap-4589	217	6	g2	g2	PROPN
ap-4589	217	7	j2	j2	PROPN
ap-4589	217	8	3	3	X
ap-4589	217	9	=	=	SYM
ap-4589	217	10	j+j−	j+j−	PROPN
ap-4589	217	11	−	−	PROPN
ap-4589	217	12	(	(	PUNCT
ap-4589	217	13	j3	j3	PROPN
ap-4589	217	14	−	−	PROPN
ap-4589	217	15	1)2	1)2	PROPN
ap-4589	217	16	−	−	PROPN
ap-4589	218	1	g2	g2	PROPN
ap-4589	218	2	(	(	PUNCT
ap-4589	218	3	j3	j3	PROPN
ap-4589	218	4	−	−	PROPN
ap-4589	218	5	1)2	1)2	NUM
ap-4589	218	6	=	=	PUNCT
ap-4589	218	7	−∂2	−∂2	NOUN
ap-4589	218	8	x	x	SYM
ap-4589	218	9	−	−	NOUN
ap-4589	218	10	j3(j3	j3(j3	ADJ
ap-4589	218	11	−	−	PROPN
ap-4589	218	12	1	1	NUM
ap-4589	218	13	)	)	PUNCT
ap-4589	218	14	cosh2	cosh2	VERB
ap-4589	218	15	x	x	PUNCT
ap-4589	219	1	−	−	PROPN
ap-4589	219	2	2	2	NUM
ap-4589	219	3	g	g	NOUN
ap-4589	219	4	tanh	tanh	PROPN
ap-4589	219	5	x.	x.	NOUN
ap-4589	219	6	(	(	PUNCT
ap-4589	219	7	4.18	4.18	NUM
ap-4589	219	8	)	)	PUNCT
ap-4589	219	9	note	note	NOUN
ap-4589	219	10	that	that	SCONJ
ap-4589	219	11	the	the	DET
ap-4589	219	12	commutation	commutation	NOUN
ap-4589	219	13	relations	relation	NOUN
ap-4589	219	14	(	(	PUNCT
ap-4589	219	15	4.17	4.17	NUM
ap-4589	219	16	)	)	PUNCT
ap-4589	219	17	are	be	AUX
ap-4589	219	18	exactly	exactly	ADV
ap-4589	219	19	the	the	DET
ap-4589	219	20	same	same	ADJ
ap-4589	219	21	as	as	ADP
ap-4589	219	22	those	those	PRON
ap-4589	219	23	for	for	ADP
ap-4589	219	24	the	the	DET
ap-4589	219	25	hyperbolic	hyperbolic	ADJ
ap-4589	219	26	kepler	kepler	PROPN
ap-4589	219	27	problem	problem	NOUN
ap-4589	219	28	.	.	PUNCT
ap-4589	220	1	hence	hence	ADV
ap-4589	220	2	the	the	DET
ap-4589	220	3	bound	bind	VERB
ap-4589	220	4	-	-	PUNCT
ap-4589	220	5	state	state	NOUN
ap-4589	220	6	spectrum	spectrum	NOUN
ap-4589	220	7	should	should	AUX
ap-4589	220	8	be	be	AUX
ap-4589	220	9	related	relate	VERB
ap-4589	220	10	to	to	ADP
ap-4589	220	11	the	the	DET
ap-4589	220	12	representations	representation	NOUN
ap-4589	220	13	classified	classify	VERB
ap-4589	220	14	in	in	ADP
ap-4589	220	15	the	the	DET
ap-4589	220	16	previous	previous	ADJ
ap-4589	220	17	subsection	subsection	NOUN
ap-4589	220	18	.	.	PUNCT
ap-4589	221	1	5this	5this	NUM
ap-4589	221	2	is	be	AUX
ap-4589	221	3	,	,	PUNCT
ap-4589	221	4	of	of	ADP
ap-4589	221	5	course	course	NOUN
ap-4589	221	6	,	,	PUNCT
ap-4589	221	7	not	not	PART
ap-4589	221	8	sufficient	sufficient	ADJ
ap-4589	221	9	condition	condition	NOUN
ap-4589	221	10	.	.	PUNCT
ap-4589	222	1	452	452	NUM
ap-4589	222	2	vol	vol	NOUN
ap-4589	222	3	.	.	PUNCT
ap-4589	222	4	57	57	NUM
ap-4589	222	5	no	no	NOUN
ap-4589	222	6	.	.	PUNCT
ap-4589	223	1	6/2017	6/2017	X
ap-4589	223	2	algebraic	algebraic	ADJ
ap-4589	223	3	description	description	NOUN
ap-4589	223	4	of	of	ADP
ap-4589	223	5	shape	shape	NOUN
ap-4589	223	6	invariance	invariance	NOUN
ap-4589	223	7	revisited	revisit	VERB
ap-4589	223	8	to	to	PART
ap-4589	223	9	understand	understand	VERB
ap-4589	223	10	which	which	DET
ap-4589	223	11	representations	representation	NOUN
ap-4589	223	12	are	be	AUX
ap-4589	223	13	realized	realize	VERB
ap-4589	223	14	,	,	PUNCT
ap-4589	223	15	let	let	VERB
ap-4589	223	16	us	we	PRON
ap-4589	223	17	study	study	VERB
ap-4589	223	18	the	the	DET
ap-4589	223	19	minimum	minimum	NOUN
ap-4589	223	20	of	of	ADP
ap-4589	223	21	the	the	DET
ap-4589	223	22	potential	potential	ADJ
ap-4589	223	23	v	v	NOUN
ap-4589	223	24	(	(	PUNCT
ap-4589	223	25	x	x	NOUN
ap-4589	223	26	)	)	PUNCT
ap-4589	223	27	=	=	SYM
ap-4589	223	28	−j(j	−j(j	NOUN
ap-4589	224	1	−	−	PROPN
ap-4589	224	2	1)/	1)/	NUM
ap-4589	224	3	cosh2	cosh2	VERB
ap-4589	224	4	x−	x−	PROPN
ap-4589	224	5	2	2	NUM
ap-4589	224	6	g	g	NOUN
ap-4589	224	7	tanh	tanh	PROPN
ap-4589	224	8	x.	x.	NOUN
ap-4589	224	9	thanks	thank	NOUN
ap-4589	224	10	to	to	ADP
ap-4589	224	11	the	the	DET
ap-4589	224	12	symmetry	symmetry	NOUN
ap-4589	224	13	j	j	PROPN
ap-4589	224	14	→	→	SYM
ap-4589	224	15	1−	1−	NUM
ap-4589	224	16	j	j	PROPN
ap-4589	224	17	,	,	PUNCT
ap-4589	224	18	without	without	ADP
ap-4589	224	19	any	any	DET
ap-4589	224	20	loss	loss	NOUN
ap-4589	224	21	of	of	ADP
ap-4589	224	22	generality	generality	NOUN
ap-4589	224	23	we	we	PRON
ap-4589	224	24	can	can	AUX
ap-4589	224	25	focus	focus	VERB
ap-4589	224	26	on	on	ADP
ap-4589	224	27	the	the	DET
ap-4589	224	28	case	case	NOUN
ap-4589	224	29	j	j	PROPN
ap-4589	224	30	≥	≥	NUM
ap-4589	224	31	1/2	1/2	NUM
ap-4589	224	32	.	.	PUNCT
ap-4589	225	1	it	it	PRON
ap-4589	225	2	is	be	AUX
ap-4589	225	3	then	then	ADV
ap-4589	225	4	easy	easy	ADJ
ap-4589	225	5	to	to	PART
ap-4589	225	6	see	see	VERB
ap-4589	225	7	that	that	SCONJ
ap-4589	225	8	the	the	DET
ap-4589	225	9	potential	potential	NOUN
ap-4589	225	10	has	have	VERB
ap-4589	225	11	a	a	DET
ap-4589	225	12	minimum	minimum	NOUN
ap-4589	225	13	if	if	SCONJ
ap-4589	225	14	j	j	PROPN
ap-4589	225	15	is	be	AUX
ap-4589	225	16	in	in	ADP
ap-4589	225	17	the	the	DET
ap-4589	225	18	range	range	NOUN
ap-4589	225	19	(	(	PUNCT
ap-4589	225	20	1	1	NUM
ap-4589	225	21	2	2	NUM
ap-4589	225	22	+	+	CCONJ
ap-4589	225	23	√	√	PROPN
ap-4589	225	24	g	g	NOUN
ap-4589	225	25	+	+	CCONJ
ap-4589	225	26	1	1	NUM
ap-4589	225	27	4	4	NUM
ap-4589	225	28	,	,	PUNCT
ap-4589	225	29	∞	∞	PROPN
ap-4589	225	30	)	)	PUNCT
ap-4589	225	31	,	,	PUNCT
ap-4589	225	32	which	which	PRON
ap-4589	225	33	contains	contain	VERB
ap-4589	225	34	the	the	DET
ap-4589	225	35	region	region	NOUN
ap-4589	225	36	(	(	PUNCT
ap-4589	225	37	1	1	NUM
ap-4589	225	38	+	+	NOUN
ap-4589	225	39	√g,∞	√g,∞	NOUN
ap-4589	225	40	)	)	PUNCT
ap-4589	225	41	;	;	PUNCT
ap-4589	225	42	see	see	VERB
ap-4589	225	43	figure	figure	NOUN
ap-4589	225	44	2(c	2(c	NUM
ap-4589	225	45	)	)	PUNCT
ap-4589	225	46	.	.	PUNCT
ap-4589	226	1	hence	hence	ADV
ap-4589	226	2	,	,	PUNCT
ap-4589	226	3	in	in	ADP
ap-4589	226	4	contrast	contrast	NOUN
ap-4589	226	5	to	to	ADP
ap-4589	226	6	the	the	DET
ap-4589	226	7	previous	previous	ADJ
ap-4589	226	8	case	case	NOUN
ap-4589	226	9	,	,	PUNCT
ap-4589	226	10	the	the	DET
ap-4589	226	11	bound	bind	VERB
ap-4589	226	12	-	-	PUNCT
ap-4589	226	13	state	state	NOUN
ap-4589	226	14	problem	problem	NOUN
ap-4589	226	15	for	for	ADP
ap-4589	226	16	the	the	DET
ap-4589	226	17	rosen	rosen	PROPN
ap-4589	226	18	–	–	PUNCT
ap-4589	226	19	morse	morse	ADJ
ap-4589	226	20	hamiltonian	hamiltonian	NOUN
ap-4589	226	21	should	should	AUX
ap-4589	226	22	be	be	AUX
ap-4589	226	23	related	relate	VERB
ap-4589	226	24	to	to	ADP
ap-4589	226	25	the	the	DET
ap-4589	226	26	infinite	infinite	ADJ
ap-4589	226	27	-	-	PUNCT
ap-4589	226	28	dimensional	dimensional	ADJ
ap-4589	226	29	representation	representation	NOUN
ap-4589	226	30	(	(	PUNCT
ap-4589	226	31	4.10	4.10	NUM
ap-4589	226	32	)	)	PUNCT
ap-4589	226	33	.	.	PUNCT
ap-4589	227	1	now	now	ADV
ap-4589	227	2	it	it	PRON
ap-4589	227	3	is	be	AUX
ap-4589	227	4	easy	easy	ADJ
ap-4589	227	5	to	to	PART
ap-4589	227	6	find	find	VERB
ap-4589	227	7	the	the	DET
ap-4589	227	8	energy	energy	NOUN
ap-4589	227	9	eigenvalue	eigenvalue	NOUN
ap-4589	227	10	of	of	ADP
ap-4589	227	11	the	the	DET
ap-4589	227	12	original	original	ADJ
ap-4589	227	13	hamiltonian	hamiltonian	NOUN
ap-4589	227	14	.	.	PUNCT
ap-4589	228	1	let	let	VERB
ap-4589	228	2	j	j	PROPN
ap-4589	228	3	∈	∈	PROPN
ap-4589	228	4	(	(	PUNCT
ap-4589	228	5	1	1	NUM
ap-4589	228	6	+	+	NOUN
ap-4589	228	7	√g,∞	√g,∞	NOUN
ap-4589	228	8	)	)	PUNCT
ap-4589	228	9	be	be	AUX
ap-4589	228	10	fixed	fix	VERB
ap-4589	228	11	.	.	PUNCT
ap-4589	229	1	then	then	ADV
ap-4589	229	2	the	the	DET
ap-4589	229	3	energy	energy	NOUN
ap-4589	229	4	eigenvalues	eigenvalue	NOUN
ap-4589	229	5	and	and	CCONJ
ap-4589	229	6	eigenfunctions	eigenfunction	NOUN
ap-4589	229	7	read	read	VERB
ap-4589	229	8	en	en	X
ap-4589	229	9	=	=	PUNCT
ap-4589	229	10	−(j	−(j	NOUN
ap-4589	229	11	−	−	PROPN
ap-4589	229	12	n−	n−	NOUN
ap-4589	229	13	1)2	1)2	NUM
ap-4589	229	14	−	−	NUM
ap-4589	229	15	g2	g2	PROPN
ap-4589	229	16	(	(	PUNCT
ap-4589	229	17	j	j	PROPN
ap-4589	229	18	−	−	PROPN
ap-4589	229	19	n−	n−	PROPN
ap-4589	229	20	1)2	1)2	NUM
ap-4589	229	21	,	,	PUNCT
ap-4589	229	22	n	n	CCONJ
ap-4589	229	23	∈	∈	PROPN
ap-4589	229	24	{	{	PUNCT
ap-4589	229	25	0	0	NUM
ap-4589	229	26	,	,	PUNCT
ap-4589	229	27	1	1	NUM
ap-4589	229	28	,	,	PUNCT
ap-4589	229	29	·	·	PUNCT
ap-4589	229	30	·	·	PUNCT
ap-4589	229	31	·	·	PUNCT
ap-4589	229	32	,	,	PUNCT
ap-4589	229	33	n	n	CCONJ
ap-4589	229	34	}	}	PUNCT
ap-4589	229	35	,	,	PUNCT
ap-4589	229	36	(	(	PUNCT
ap-4589	229	37	4.19	4.19	NUM
ap-4589	229	38	)	)	PUNCT
ap-4589	229	39	and	and	CCONJ
ap-4589	229	40	ψen	ψen	NOUN
ap-4589	229	41	,	,	PUNCT
ap-4589	229	42	j(x	j(x	PROPN
ap-4589	229	43	)	)	PUNCT
ap-4589	229	44	∝	∝	PROPN
ap-4589	229	45	a+(j	a+(j	ADV
ap-4589	230	1	−	−	PROPN
ap-4589	230	2	1)a+(j	1)a+(j	NUM
ap-4589	230	3	−	−	NOUN
ap-4589	230	4	2	2	NUM
ap-4589	230	5	)	)	PUNCT
ap-4589	230	6	·	·	PUNCT
ap-4589	230	7	·	·	PUNCT
ap-4589	230	8	·	·	PUNCT
ap-4589	230	9	a+(j	a+(j	NOUN
ap-4589	231	1	−	−	ADP
ap-4589	231	2	n)ψen	n)ψen	NOUN
ap-4589	231	3	,	,	PUNCT
ap-4589	231	4	j−n(x	j−n(x	ADJ
ap-4589	231	5	)	)	PUNCT
ap-4589	231	6	,	,	PUNCT
ap-4589	231	7	(	(	PUNCT
ap-4589	231	8	4.20	4.20	NUM
ap-4589	231	9	)	)	PUNCT
ap-4589	231	10	where	where	SCONJ
ap-4589	231	11	n	n	NOUN
ap-4589	231	12	=	=	SYM
ap-4589	231	13	max{n	max{n	CCONJ
ap-4589	231	14	∈	∈	PROPN
ap-4589	231	15	z≥0	z≥0	NOUN
ap-4589	231	16	:	:	PUNCT
ap-4589	231	17	1	1	NUM
ap-4589	231	18	+	+	NOUN
ap-4589	231	19	√g	√g	X
ap-4589	231	20	<	<	X
ap-4589	231	21	j	j	PROPN
ap-4589	231	22	−	−	PROPN
ap-4589	231	23	n	n	CCONJ
ap-4589	231	24	}	}	PUNCT
ap-4589	231	25	and	and	CCONJ
ap-4589	231	26	ψen	ψen	NOUN
ap-4589	231	27	,	,	PUNCT
ap-4589	231	28	j−n(x	j−n(x	ADJ
ap-4589	231	29	)	)	PUNCT
ap-4589	231	30	∝	∝	PROPN
ap-4589	231	31	(	(	PUNCT
ap-4589	231	32	cosh	cosh	PROPN
ap-4589	231	33	x)−j+n+1	x)−j+n+1	PROPN
ap-4589	231	34	exp	exp	PROPN
ap-4589	231	35	(	(	PUNCT
ap-4589	231	36	g	g	NOUN
ap-4589	231	37	j−n−1x	j−n−1x	NOUN
ap-4589	231	38	)	)	PUNCT
ap-4589	231	39	.	.	PUNCT
ap-4589	232	1	notice	notice	VERB
ap-4589	232	2	that	that	SCONJ
ap-4589	232	3	(	(	PUNCT
ap-4589	232	4	4.19	4.19	NUM
ap-4589	232	5	)	)	PUNCT
ap-4589	232	6	and	and	CCONJ
ap-4589	232	7	(	(	PUNCT
ap-4589	232	8	4.20	4.20	NUM
ap-4589	232	9	)	)	PUNCT
ap-4589	232	10	are	be	AUX
ap-4589	232	11	consistent	consistent	ADJ
ap-4589	232	12	with	with	ADP
ap-4589	232	13	the	the	DET
ap-4589	232	14	known	know	VERB
ap-4589	232	15	results	result	NOUN
ap-4589	232	16	[	[	X
ap-4589	232	17	7	7	NUM
ap-4589	232	18	]	]	PUNCT
ap-4589	232	19	.	.	PUNCT
ap-4589	233	1	5	5	X
ap-4589	233	2	.	.	X
ap-4589	233	3	conclusions	conclusion	NOUN
ap-4589	233	4	in	in	ADP
ap-4589	233	5	this	this	DET
ap-4589	233	6	note	note	NOUN
ap-4589	233	7	we	we	PRON
ap-4589	233	8	have	have	AUX
ap-4589	233	9	revisited	revisit	VERB
ap-4589	233	10	the	the	DET
ap-4589	233	11	bound	bind	VERB
ap-4589	233	12	-	-	PUNCT
ap-4589	233	13	state	state	NOUN
ap-4589	233	14	problems	problem	NOUN
ap-4589	233	15	for	for	ADP
ap-4589	233	16	the	the	DET
ap-4589	233	17	kepler	kepler	NOUN
ap-4589	233	18	,	,	PUNCT
ap-4589	233	19	spherical	spherical	ADJ
ap-4589	233	20	kepler	kepler	NOUN
ap-4589	233	21	,	,	PUNCT
ap-4589	233	22	hyperbolic	hyperbolic	ADJ
ap-4589	233	23	kepler	kepler	NOUN
ap-4589	233	24	,	,	PUNCT
ap-4589	233	25	and	and	CCONJ
ap-4589	233	26	rosen	rosen	PROPN
ap-4589	233	27	–	–	PUNCT
ap-4589	233	28	morse	morse	PROPN
ap-4589	233	29	hamiltonians	hamiltonian	NOUN
ap-4589	233	30	,	,	PUNCT
ap-4589	233	31	all	all	PRON
ap-4589	233	32	of	of	ADP
ap-4589	233	33	which	which	PRON
ap-4589	233	34	have	have	AUX
ap-4589	233	35	not	not	PART
ap-4589	233	36	been	be	AUX
ap-4589	233	37	solved	solve	VERB
ap-4589	233	38	before	before	ADV
ap-4589	233	39	in	in	ADP
ap-4589	233	40	terms	term	NOUN
ap-4589	233	41	of	of	ADP
ap-4589	233	42	potential	potential	ADJ
ap-4589	233	43	algebra	algebra	NOUN
ap-4589	233	44	.	.	PUNCT
ap-4589	234	1	we	we	PRON
ap-4589	234	2	have	have	AUX
ap-4589	234	3	introduced	introduce	VERB
ap-4589	234	4	three	three	NUM
ap-4589	234	5	nonlinear	nonlinear	ADJ
ap-4589	234	6	algebraic	algebraic	ADJ
ap-4589	234	7	systems	system	NOUN
ap-4589	234	8	and	and	CCONJ
ap-4589	234	9	solved	solve	VERB
ap-4589	234	10	the	the	DET
ap-4589	234	11	problems	problem	NOUN
ap-4589	234	12	by	by	ADP
ap-4589	234	13	means	mean	NOUN
ap-4589	234	14	of	of	ADP
ap-4589	234	15	representation	representation	NOUN
ap-4589	234	16	theory	theory	NOUN
ap-4589	234	17	.	.	PUNCT
ap-4589	235	1	we	we	PRON
ap-4589	235	2	have	have	AUX
ap-4589	235	3	seen	see	VERB
ap-4589	235	4	that	that	SCONJ
ap-4589	235	5	the	the	DET
ap-4589	235	6	discrete	discrete	ADJ
ap-4589	235	7	energy	energy	NOUN
ap-4589	235	8	spectra	spectra	NOUN
ap-4589	235	9	can	can	AUX
ap-4589	235	10	be	be	AUX
ap-4589	235	11	obtained	obtain	VERB
ap-4589	235	12	just	just	ADV
ap-4589	235	13	from	from	ADP
ap-4589	235	14	the	the	DET
ap-4589	235	15	four	four	NUM
ap-4589	235	16	conditions	condition	NOUN
ap-4589	235	17	:	:	PUNCT
ap-4589	235	18	j±|e	j±|e	ADJ
ap-4589	235	19	,	,	PUNCT
ap-4589	235	20	j	j	PROPN
ap-4589	235	21	〉	〉	PROPN
ap-4589	235	22	∝	∝	PROPN
ap-4589	235	23	|e	|e	PROPN
ap-4589	235	24	,	,	PUNCT
ap-4589	235	25	j	j	PROPN
ap-4589	235	26	±	±	PROPN
ap-4589	235	27	1	1	NUM
ap-4589	235	28	〉	〉	NUM
ap-4589	235	29	and	and	CCONJ
ap-4589	235	30	‖j±|e	‖j±|e	PROPN
ap-4589	235	31	,	,	PUNCT
ap-4589	235	32	j〉‖2	j〉‖2	ADJ
ap-4589	235	33	≥	≥	NOUN
ap-4589	235	34	0	0	NUM
ap-4589	235	35	.	.	PUNCT
ap-4589	236	1	these	these	DET
ap-4589	236	2	conditions	condition	NOUN
ap-4589	236	3	correctly	correctly	ADV
ap-4589	236	4	reproduce	reproduce	VERB
ap-4589	236	5	the	the	DET
ap-4589	236	6	known	know	VERB
ap-4589	236	7	results	result	NOUN
ap-4589	236	8	in	in	ADP
ap-4589	236	9	a	a	DET
ap-4589	236	10	purely	purely	ADV
ap-4589	236	11	algebraic	algebraic	ADJ
ap-4589	236	12	fashion	fashion	NOUN
ap-4589	236	13	.	.	PUNCT
ap-4589	237	1	the	the	DET
ap-4589	237	2	price	price	NOUN
ap-4589	237	3	to	to	PART
ap-4589	237	4	pay	pay	VERB
ap-4589	237	5	,	,	PUNCT
ap-4589	237	6	however	however	ADV
ap-4589	237	7	,	,	PUNCT
ap-4589	237	8	is	be	AUX
ap-4589	237	9	that	that	SCONJ
ap-4589	237	10	in	in	ADP
ap-4589	237	11	this	this	DET
ap-4589	237	12	approach	approach	NOUN
ap-4589	237	13	j	j	PROPN
ap-4589	237	14	must	must	AUX
ap-4589	237	15	be	be	AUX
ap-4589	237	16	a	a	DET
ap-4589	237	17	half	half	ADJ
ap-4589	237	18	-	-	PUNCT
ap-4589	237	19	integer	integer	NOUN
ap-4589	237	20	(	(	PUNCT
ap-4589	237	21	except	except	SCONJ
ap-4589	237	22	for	for	ADP
ap-4589	237	23	the	the	DET
ap-4589	237	24	rosen	rosen	PROPN
ap-4589	237	25	–	–	PUNCT
ap-4589	237	26	morse	morse	ADJ
ap-4589	237	27	potential	potential	ADJ
ap-4589	237	28	problem	problem	NOUN
ap-4589	237	29	and	and	CCONJ
ap-4589	237	30	the	the	DET
ap-4589	237	31	hyperbolic	hyperbolic	ADJ
ap-4589	237	32	kepler	kepler	NOUN
ap-4589	237	33	problem	problem	NOUN
ap-4589	237	34	in	in	ADP
ap-4589	237	35	the	the	DET
ap-4589	237	36	domain	domain	NOUN
ap-4589	237	37	g	g	PROPN
ap-4589	237	38	∈	∈	PROPN
ap-4589	237	39	(	(	PUNCT
ap-4589	237	40	0	0	NUM
ap-4589	237	41	,	,	PUNCT
ap-4589	237	42	1/4	1/4	NUM
ap-4589	237	43	)	)	PUNCT
ap-4589	237	44	)	)	PUNCT
ap-4589	237	45	,	,	PUNCT
ap-4589	237	46	otherwise	otherwise	ADV
ap-4589	237	47	there	there	PRON
ap-4589	237	48	arise	arise	VERB
ap-4589	237	49	inconsistencies	inconsistency	NOUN
ap-4589	237	50	.	.	PUNCT
ap-4589	238	1	this	this	PRON
ap-4589	238	2	is	be	AUX
ap-4589	238	3	a	a	DET
ap-4589	238	4	weakness	weakness	NOUN
ap-4589	238	5	of	of	ADP
ap-4589	238	6	this	this	DET
ap-4589	238	7	representation	representation	NOUN
ap-4589	238	8	theoretic	theoretic	NOUN
ap-4589	238	9	approach	approach	NOUN
ap-4589	238	10	.	.	PUNCT
ap-4589	239	1	references	reference	NOUN
ap-4589	239	2	[	[	X
ap-4589	239	3	1	1	X
ap-4589	239	4	]	]	PUNCT
ap-4589	239	5	e.	e.	PROPN
ap-4589	239	6	schrödinger	schrödinger	PROPN
ap-4589	239	7	.	.	PUNCT
ap-4589	240	1	a	a	DET
ap-4589	240	2	method	method	NOUN
ap-4589	240	3	of	of	ADP
ap-4589	240	4	determining	determine	VERB
ap-4589	240	5	quantum	quantum	ADJ
ap-4589	240	6	-	-	ADJ
ap-4589	240	7	mechanical	mechanical	ADJ
ap-4589	240	8	eigenvalues	eigenvalue	NOUN
ap-4589	240	9	and	and	CCONJ
ap-4589	240	10	eigenfunctions	eigenfunction	NOUN
ap-4589	240	11	.	.	PUNCT
ap-4589	241	1	proc	proc	PROPN
ap-4589	241	2	roy	roy	PROPN
ap-4589	241	3	irish	irish	PROPN
ap-4589	241	4	acad	acad	PROPN
ap-4589	241	5	(	(	PUNCT
ap-4589	241	6	sect	sect	NOUN
ap-4589	241	7	a	a	X
ap-4589	241	8	)	)	PUNCT
ap-4589	241	9	46:9–16	46:9–16	NUM
ap-4589	241	10	,	,	PUNCT
ap-4589	241	11	1940	1940	NUM
ap-4589	241	12	.	.	PUNCT
ap-4589	242	1	[	[	X
ap-4589	242	2	2	2	NUM
ap-4589	242	3	]	]	X
ap-4589	242	4	l.	l.	PROPN
ap-4589	242	5	infeld	infeld	PROPN
ap-4589	242	6	.	.	PUNCT
ap-4589	243	1	on	on	ADP
ap-4589	243	2	a	a	DET
ap-4589	243	3	new	new	ADJ
ap-4589	243	4	treatment	treatment	NOUN
ap-4589	243	5	of	of	ADP
ap-4589	243	6	some	some	DET
ap-4589	243	7	eigenvalue	eigenvalue	ADJ
ap-4589	243	8	problems	problem	NOUN
ap-4589	243	9	.	.	PUNCT
ap-4589	244	1	phys	phy	NOUN
ap-4589	244	2	rev	rev	VERB
ap-4589	244	3	59:737–747	59:737–747	PROPN
ap-4589	244	4	,	,	PUNCT
ap-4589	244	5	1941	1941	NUM
ap-4589	244	6	.	.	PUNCT
ap-4589	245	1	doi:10.1103	doi:10.1103	NOUN
ap-4589	245	2	/	/	SYM
ap-4589	245	3	physrev.59.737	physrev.59.737	VERB
ap-4589	245	4	.	.	PUNCT
ap-4589	246	1	[	[	X
ap-4589	246	2	3	3	NUM
ap-4589	246	3	]	]	PUNCT
ap-4589	246	4	a.	a.	PROPN
ap-4589	246	5	f.	f.	PROPN
ap-4589	246	6	stevenson	stevenson	PROPN
ap-4589	246	7	.	.	PUNCT
ap-4589	247	1	note	note	NOUN
ap-4589	247	2	on	on	ADP
ap-4589	247	3	the	the	DET
ap-4589	247	4	“	"	PUNCT
ap-4589	247	5	kepler	kepler	NOUN
ap-4589	247	6	problem	problem	NOUN
ap-4589	247	7	”	"	PUNCT
ap-4589	247	8	in	in	ADP
ap-4589	247	9	a	a	DET
ap-4589	247	10	spherical	spherical	ADJ
ap-4589	247	11	space	space	NOUN
ap-4589	247	12	,	,	PUNCT
ap-4589	247	13	and	and	CCONJ
ap-4589	247	14	the	the	DET
ap-4589	247	15	factorization	factorization	NOUN
ap-4589	247	16	method	method	NOUN
ap-4589	247	17	of	of	ADP
ap-4589	247	18	solving	solve	VERB
ap-4589	247	19	eigenvalue	eigenvalue	NOUN
ap-4589	247	20	problems	problem	NOUN
ap-4589	247	21	.	.	PUNCT
ap-4589	248	1	phys	phy	NOUN
ap-4589	248	2	rev	rev	VERB
ap-4589	248	3	59:842–843	59:842–843	PROPN
ap-4589	248	4	,	,	PUNCT
ap-4589	248	5	1941	1941	NUM
ap-4589	248	6	.	.	PUNCT
ap-4589	249	1	doi:10.1103	doi:10.1103	NOUN
ap-4589	249	2	/	/	SYM
ap-4589	249	3	physrev.59.842	physrev.59.842	VERB
ap-4589	249	4	.	.	PUNCT
ap-4589	250	1	[	[	X
ap-4589	250	2	4	4	X
ap-4589	250	3	]	]	PUNCT
ap-4589	250	4	m.	m.	PROPN
ap-4589	250	5	f.	f.	PROPN
ap-4589	250	6	manning	manning	PROPN
ap-4589	250	7	,	,	PUNCT
ap-4589	250	8	n.	n.	PROPN
ap-4589	250	9	rosen	rosen	PROPN
ap-4589	250	10	.	.	PUNCT
ap-4589	251	1	a	a	DET
ap-4589	251	2	potential	potential	ADJ
ap-4589	251	3	function	function	NOUN
ap-4589	251	4	for	for	ADP
ap-4589	251	5	the	the	DET
ap-4589	251	6	vibrations	vibration	NOUN
ap-4589	251	7	of	of	ADP
ap-4589	251	8	diatomic	diatomic	ADJ
ap-4589	251	9	molecule	molecule	NOUN
ap-4589	251	10	.	.	PUNCT
ap-4589	252	1	phys	phy	NOUN
ap-4589	252	2	rev	rev	VERB
ap-4589	252	3	44:951–954	44:951–954	PROPN
ap-4589	252	4	,	,	PUNCT
ap-4589	252	5	1933	1933	NUM
ap-4589	252	6	.	.	PUNCT
ap-4589	253	1	doi:10.1103	doi:10.1103	PROPN
ap-4589	253	2	/	/	SYM
ap-4589	253	3	physrev.44.951	physrev.44.951	NOUN
ap-4589	253	4	.	.	PUNCT
ap-4589	254	1	[	[	X
ap-4589	254	2	5	5	NUM
ap-4589	254	3	]	]	X
ap-4589	254	4	l.	l.	PROPN
ap-4589	254	5	infeld	infeld	PROPN
ap-4589	254	6	,	,	PUNCT
ap-4589	254	7	a.	a.	PROPN
ap-4589	254	8	schild	schild	PROPN
ap-4589	254	9	.	.	PUNCT
ap-4589	255	1	a	a	DET
ap-4589	255	2	note	note	NOUN
ap-4589	255	3	on	on	ADP
ap-4589	255	4	the	the	DET
ap-4589	255	5	kepler	kepler	PROPN
ap-4589	255	6	problem	problem	NOUN
ap-4589	255	7	in	in	ADP
ap-4589	255	8	a	a	DET
ap-4589	255	9	space	space	NOUN
ap-4589	255	10	of	of	ADP
ap-4589	255	11	constant	constant	ADJ
ap-4589	255	12	negative	negative	ADJ
ap-4589	255	13	curvature	curvature	NOUN
ap-4589	255	14	.	.	PUNCT
ap-4589	256	1	phys	phy	NOUN
ap-4589	256	2	rev	rev	VERB
ap-4589	256	3	67:121–122	67:121–122	PROPN
ap-4589	256	4	,	,	PUNCT
ap-4589	256	5	1945	1945	NUM
ap-4589	256	6	.	.	PUNCT
ap-4589	257	1	doi:10.1103	doi:10.1103	SYM
ap-4589	257	2	/	/	SYM
ap-4589	257	3	physrev.67.121	physrev.67.121	NOUN
ap-4589	257	4	.	.	PUNCT
ap-4589	258	1	[	[	X
ap-4589	258	2	6	6	NUM
ap-4589	258	3	]	]	PUNCT
ap-4589	258	4	c.	c.	PROPN
ap-4589	258	5	eckart	eckart	PROPN
ap-4589	258	6	.	.	PUNCT
ap-4589	259	1	the	the	DET
ap-4589	259	2	penetration	penetration	NOUN
ap-4589	259	3	of	of	ADP
ap-4589	259	4	a	a	DET
ap-4589	259	5	potential	potential	ADJ
ap-4589	259	6	barrier	barrier	NOUN
ap-4589	259	7	by	by	ADP
ap-4589	259	8	electrons	electron	NOUN
ap-4589	259	9	.	.	PUNCT
ap-4589	260	1	phys	phy	NOUN
ap-4589	260	2	rev	rev	VERB
ap-4589	260	3	35:1303–1309	35:1303–1309	PROPN
ap-4589	260	4	,	,	PUNCT
ap-4589	260	5	1930	1930	NUM
ap-4589	260	6	.	.	PUNCT
ap-4589	261	1	doi:10.1103	doi:10.1103	NOUN
ap-4589	261	2	/	/	SYM
ap-4589	261	3	physrev.35.1303	physrev.35.1303	ADJ
ap-4589	261	4	.	.	PUNCT
ap-4589	262	1	[	[	X
ap-4589	262	2	7	7	X
ap-4589	262	3	]	]	X
ap-4589	262	4	n.	n.	PROPN
ap-4589	262	5	rosen	rosen	PROPN
ap-4589	262	6	,	,	PUNCT
ap-4589	262	7	p.	p.	PROPN
ap-4589	262	8	m.	m.	NOUN
ap-4589	262	9	morse	morse	PROPN
ap-4589	262	10	.	.	PUNCT
ap-4589	263	1	on	on	ADP
ap-4589	263	2	the	the	DET
ap-4589	263	3	vibrations	vibration	NOUN
ap-4589	263	4	of	of	ADP
ap-4589	263	5	polyatomic	polyatomic	ADJ
ap-4589	263	6	molecules	molecule	NOUN
ap-4589	263	7	.	.	PUNCT
ap-4589	264	1	phys	phy	NOUN
ap-4589	264	2	rev	rev	VERB
ap-4589	264	3	42:210–217	42:210–217	PROPN
ap-4589	264	4	,	,	PUNCT
ap-4589	264	5	1932	1932	NUM
ap-4589	264	6	.	.	PUNCT
ap-4589	265	1	doi:10.1103	doi:10.1103	NOUN
ap-4589	265	2	/	/	SYM
ap-4589	265	3	physrev.42.210	physrev.42.210	NOUN
ap-4589	265	4	.	.	PUNCT
ap-4589	266	1	[	[	X
ap-4589	266	2	8	8	NUM
ap-4589	266	3	]	]	X
ap-4589	266	4	l.	l.	PROPN
ap-4589	266	5	infeld	infeld	PROPN
ap-4589	266	6	,	,	PUNCT
ap-4589	266	7	t.	t.	PROPN
ap-4589	266	8	e.	e.	PROPN
ap-4589	266	9	hull	hull	PROPN
ap-4589	266	10	.	.	PUNCT
ap-4589	267	1	the	the	DET
ap-4589	267	2	factorization	factorization	NOUN
ap-4589	267	3	method	method	NOUN
ap-4589	267	4	.	.	PUNCT
ap-4589	268	1	rev	rev	VERB
ap-4589	268	2	mod	mod	PROPN
ap-4589	268	3	phys	phys	PROPN
ap-4589	268	4	23:21–68	23:21–68	PROPN
ap-4589	268	5	,	,	PUNCT
ap-4589	268	6	1951	1951	NUM
ap-4589	268	7	.	.	PUNCT
ap-4589	269	1	doi:10.1103	doi:10.1103	NOUN
ap-4589	269	2	/	/	SYM
ap-4589	269	3	revmodphys.23.21	revmodphys.23.21	PROPN
ap-4589	269	4	.	.	PUNCT
ap-4589	270	1	[	[	X
ap-4589	270	2	9	9	NUM
ap-4589	270	3	]	]	X
ap-4589	270	4	f.	f.	PROPN
ap-4589	270	5	cooper	cooper	PROPN
ap-4589	270	6	,	,	PUNCT
ap-4589	270	7	a.	a.	PROPN
ap-4589	270	8	khare	khare	PROPN
ap-4589	270	9	,	,	PUNCT
ap-4589	270	10	u.	u.	PROPN
ap-4589	270	11	sukhatme	sukhatme	PROPN
ap-4589	270	12	.	.	PUNCT
ap-4589	271	1	supersymmetry	supersymmetry	NOUN
ap-4589	271	2	and	and	CCONJ
ap-4589	271	3	quantum	quantum	NOUN
ap-4589	271	4	mechanics	mechanic	NOUN
ap-4589	271	5	.	.	PUNCT
ap-4589	272	1	phys	phy	NOUN
ap-4589	272	2	rept	rept	VERB
ap-4589	272	3	251:267–385	251:267–385	NUM
ap-4589	272	4	,	,	PUNCT
ap-4589	272	5	1995	1995	NUM
ap-4589	272	6	.	.	PUNCT
ap-4589	273	1	arxiv	arxiv	NOUN
ap-4589	273	2	:	:	PUNCT
ap-4589	273	3	hep	hep	PROPN
ap-4589	273	4	-	-	ADJ
ap-4589	273	5	th/9405029	th/9405029	PROPN
ap-4589	273	6	doi:10.1016/0370	doi:10.1016/0370	NOUN
ap-4589	273	7	-	-	PUNCT
ap-4589	273	8	1573(94)00080	1573(94)00080	NOUN
ap-4589	273	9	-	-	PUNCT
ap-4589	273	10	m.	m.	NOUN
ap-4589	273	11	[	[	X
ap-4589	273	12	10	10	NUM
ap-4589	273	13	]	]	PUNCT
ap-4589	273	14	t.	t.	PROPN
ap-4589	273	15	houri	houri	PROPN
ap-4589	273	16	,	,	PUNCT
ap-4589	273	17	m.	m.	NOUN
ap-4589	273	18	sakamoto	sakamoto	PROPN
ap-4589	273	19	,	,	PUNCT
ap-4589	273	20	k.	k.	PROPN
ap-4589	273	21	tatsumi	tatsumi	PROPN
ap-4589	273	22	.	.	PUNCT
ap-4589	274	1	spectral	spectral	ADJ
ap-4589	274	2	intertwining	intertwine	VERB
ap-4589	274	3	relations	relation	NOUN
ap-4589	274	4	in	in	ADP
ap-4589	274	5	exactly	exactly	ADV
ap-4589	274	6	solvable	solvable	ADJ
ap-4589	274	7	quantum	quantum	ADJ
ap-4589	274	8	-	-	ADJ
ap-4589	274	9	mechanical	mechanical	ADJ
ap-4589	274	10	systems	system	NOUN
ap-4589	274	11	.	.	PUNCT
ap-4589	275	1	ptep	ptep	NOUN
ap-4589	275	2	2017:063a01	2017:063a01	NOUN
ap-4589	275	3	,	,	PUNCT
ap-4589	275	4	2017	2017	NUM
ap-4589	275	5	.	.	PUNCT
ap-4589	276	1	arxiv:1701.04307	arxiv:1701.04307	NUM
ap-4589	276	2	doi:10.1093	doi:10.1093	NOUN
ap-4589	276	3	/	/	SYM
ap-4589	276	4	ptep	ptep	NOUN
ap-4589	276	5	/	/	SYM
ap-4589	276	6	ptx074	ptx074	PROPN
ap-4589	276	7	.	.	PUNCT
ap-4589	277	1	[	[	X
ap-4589	277	2	11	11	NUM
ap-4589	277	3	]	]	PUNCT
ap-4589	277	4	a.	a.	NOUN
ap-4589	277	5	gangopadhyaya	gangopadhyaya	PROPN
ap-4589	277	6	,	,	PUNCT
ap-4589	277	7	j.	j.	PROPN
ap-4589	277	8	v.	v.	PROPN
ap-4589	277	9	mallow	mallow	PROPN
ap-4589	277	10	,	,	PUNCT
ap-4589	277	11	u.	u.	PROPN
ap-4589	277	12	p.	p.	PROPN
ap-4589	277	13	sukhatme	sukhatme	PROPN
ap-4589	277	14	.	.	PUNCT
ap-4589	278	1	translational	translational	ADJ
ap-4589	278	2	shape	shape	NOUN
ap-4589	278	3	invariance	invariance	NOUN
ap-4589	278	4	and	and	CCONJ
ap-4589	278	5	the	the	DET
ap-4589	278	6	inherent	inherent	ADJ
ap-4589	278	7	potential	potential	ADJ
ap-4589	278	8	algebra	algebra	NOUN
ap-4589	278	9	.	.	PUNCT
ap-4589	279	1	phys	phy	NOUN
ap-4589	279	2	rev	rev	VERB
ap-4589	279	3	a58:4287–4292	a58:4287–4292	PROPN
ap-4589	279	4	,	,	PUNCT
ap-4589	279	5	1998	1998	NUM
ap-4589	279	6	.	.	PUNCT
ap-4589	280	1	doi:10.1103	doi:10.1103	NOUN
ap-4589	280	2	/	/	SYM
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ap-4589	280	4	.	.	PUNCT
ap-4589	281	1	[	[	X
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ap-4589	281	3	]	]	X
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ap-4589	281	5	rasinariu	rasinariu	PROPN
ap-4589	281	6	,	,	PUNCT
ap-4589	281	7	j.	j.	PROPN
ap-4589	281	8	v.	v.	PROPN
ap-4589	281	9	mallow	mallow	PROPN
ap-4589	281	10	,	,	PUNCT
ap-4589	281	11	a.	a.	NOUN
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ap-4589	282	2	solvable	solvable	ADJ
ap-4589	282	3	problems	problem	NOUN
ap-4589	282	4	of	of	ADP
ap-4589	282	5	quantum	quantum	ADJ
ap-4589	282	6	mechanics	mechanic	NOUN
ap-4589	282	7	and	and	CCONJ
ap-4589	282	8	their	their	PRON
ap-4589	282	9	spectrum	spectrum	NOUN
ap-4589	282	10	generating	generating	NOUN
ap-4589	282	11	algebras	algebra	NOUN
ap-4589	282	12	:	:	PUNCT
ap-4589	282	13	a	a	DET
ap-4589	282	14	review	review	NOUN
ap-4589	282	15	.	.	PUNCT
ap-4589	283	1	central	central	ADJ
ap-4589	283	2	eur	eur	PROPN
ap-4589	283	3	j	j	PROPN
ap-4589	283	4	phys	phy	NOUN
ap-4589	283	5	5:111–134	5:111–134	NUM
ap-4589	283	6	,	,	PUNCT
ap-4589	283	7	2007	2007	NUM
ap-4589	283	8	.	.	PUNCT
ap-4589	284	1	doi:10.2478	doi:10.2478	PROPN
ap-4589	284	2	/	/	SYM
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ap-4589	284	4	-	-	PUNCT
ap-4589	284	5	007	007	NUM
ap-4589	284	6	-	-	PUNCT
ap-4589	284	7	0001	0001	NUM
ap-4589	284	8	-	-	PUNCT
ap-4589	284	9	1	1	NUM
ap-4589	284	10	.	.	PUNCT
ap-4589	285	1	[	[	X
ap-4589	285	2	13	13	NUM
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ap-4589	285	5	bougie	bougie	PROPN
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ap-4589	285	13	c.	c.	PROPN
ap-4589	285	14	rasinariu	rasinariu	PROPN
ap-4589	285	15	.	.	PUNCT
ap-4589	286	1	supersymmetric	supersymmetric	ADJ
ap-4589	286	2	quantum	quantum	ADJ
ap-4589	286	3	mechanics	mechanic	NOUN
ap-4589	286	4	and	and	CCONJ
ap-4589	286	5	solvable	solvable	ADJ
ap-4589	286	6	models	model	NOUN
ap-4589	286	7	.	.	PUNCT
ap-4589	287	1	symmetry	symmetry	NOUN
ap-4589	287	2	4:452–473	4:452–473	PROPN
ap-4589	287	3	,	,	PUNCT
ap-4589	287	4	2012	2012	NUM
ap-4589	287	5	.	.	PUNCT
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ap-4589	288	2	/	/	SYM
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ap-4589	288	4	.	.	PUNCT
ap-4589	289	1	[	[	X
ap-4589	289	2	14	14	NUM
ap-4589	289	3	]	]	PUNCT
ap-4589	289	4	a.	a.	PROPN
ap-4589	289	5	b.	b.	PROPN
ap-4589	289	6	balantekin	balantekin	PROPN
ap-4589	289	7	.	.	PUNCT
ap-4589	290	1	algebraic	algebraic	ADJ
ap-4589	290	2	approach	approach	NOUN
ap-4589	290	3	to	to	PART
ap-4589	290	4	shape	shape	VERB
ap-4589	290	5	invariance	invariance	NOUN
ap-4589	290	6	.	.	PUNCT
ap-4589	291	1	phys	phy	NOUN
ap-4589	291	2	rev	rev	VERB
ap-4589	291	3	a57:4188–4191	a57:4188–4191	PROPN
ap-4589	291	4	,	,	PUNCT
ap-4589	291	5	1998	1998	NUM
ap-4589	291	6	.	.	PUNCT
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ap-4589	292	2	:	:	PUNCT
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ap-4589	292	4	-	-	PUNCT
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ap-4589	292	7	/	/	NOUN
ap-4589	292	8	physreva.57.4188	physreva.57.4188	NOUN
ap-4589	292	9	.	.	PUNCT
ap-4589	293	1	453	453	NUM
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ap-4589	293	5	http://dx.doi.org/10.1103/physrev.67.121	http://dx.doi.org/10.1103/physrev.67.121	VERB
ap-4589	293	6	http://dx.doi.org/10.1103/physrev.35.1303	http://dx.doi.org/10.1103/physrev.35.1303	PROPN
ap-4589	293	7	http://dx.doi.org/10.1103/physrev.42.210	http://dx.doi.org/10.1103/physrev.42.210	ADJ
ap-4589	293	8	http://dx.doi.org/10.1103/revmodphys.23.21	http://dx.doi.org/10.1103/revmodphys.23.21	NOUN
ap-4589	293	9	http://arxiv.org/abs/hep-th/9405029	http://arxiv.org/abs/hep-th/9405029	PROPN
ap-4589	293	10	http://dx.doi.org/10.1016/0370-1573(94)00080-m	http://dx.doi.org/10.1016/0370-1573(94)00080-m	X
ap-4589	293	11	http://arxiv.org/abs/1701.04307	http://arxiv.org/abs/1701.04307	NOUN
ap-4589	293	12	http://dx.doi.org/10.1093/ptep/ptx074	http://dx.doi.org/10.1093/ptep/ptx074	PROPN
ap-4589	293	13	http://dx.doi.org/10.1103/physreva.58.4287	http://dx.doi.org/10.1103/physreva.58.4287	PROPN
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ap-4589	294	1	http://dx.doi.org/10.3390/sym4030452	http://dx.doi.org/10.3390/sym4030452	NOUN
ap-4589	294	2	http://arxiv.org/abs/quant-ph/9712018	http://arxiv.org/abs/quant-ph/9712018	PROPN
ap-4589	294	3	http://dx.doi.org/10.1103/physreva.57.4188	http://dx.doi.org/10.1103/physreva.57.4188	PROPN
ap-4589	294	4	acta	acta	PROPN
ap-4589	294	5	polytechnica	polytechnica	PROPN
ap-4589	294	6	57(6):446–453	57(6):446–453	PROPN
ap-4589	294	7	,	,	PUNCT
ap-4589	294	8	2017	2017	NUM
ap-4589	294	9	1	1	NUM
ap-4589	294	10	introduction	introduction	NOUN
ap-4589	294	11	2	2	NUM
ap-4589	294	12	kepler	kepler	NOUN
ap-4589	294	13	3	3	NUM
ap-4589	294	14	spherical	spherical	ADJ
ap-4589	294	15	kepler	kepler	NOUN
ap-4589	294	16	4	4	NUM
ap-4589	294	17	hyperbolic	hyperbolic	ADJ
ap-4589	294	18	kepler	kepler	PROPN
ap-4589	294	19	&	&	CCONJ
ap-4589	294	20	rosen	rosen	PROPN
ap-4589	294	21	–	–	PUNCT
ap-4589	294	22	morse	morse	VERB
ap-4589	294	23	4.1	4.1	NUM
ap-4589	294	24	hyperbolic	hyperbolic	ADJ
ap-4589	294	25	kepler	kepler	PROPN
ap-4589	294	26	4.2	4.2	NUM
ap-4589	294	27	rosen	rosen	PROPN
ap-4589	294	28	–	–	PUNCT
ap-4589	294	29	morse	morse	NOUN
ap-4589	294	30	5	5	NUM
ap-4589	294	31	conclusions	conclusion	NOUN
ap-4589	294	32	references	reference	NOUN