id sid tid token lemma pos ap-2081 1 1 acta acta PROPN ap-2081 1 2 polytechnica polytechnica PROPN ap-2081 1 3 doi:10.14311 doi:10.14311 PROPN ap-2081 1 4 / / SYM ap-2081 1 5 ap.2014.54.0113 ap.2014.54.0113 PROPN ap-2081 1 6 acta acta PROPN ap-2081 1 7 polytechnica polytechnica PROPN ap-2081 1 8 54(2):113–115 54(2):113–115 PROPN ap-2081 1 9 , , PUNCT ap-2081 1 10 2014 2014 NUM ap-2081 1 11 © © PROPN ap-2081 1 12 czech czech PROPN ap-2081 1 13 technical technical PROPN ap-2081 1 14 university university PROPN ap-2081 1 15 in in ADP ap-2081 1 16 prague prague PROPN ap-2081 1 17 , , PUNCT ap-2081 1 18 2014 2014 NUM ap-2081 1 19 available available ADJ ap-2081 1 20 online online ADV ap-2081 1 21 at at ADP ap-2081 1 22 http://ojs.cvut.cz/ojs/index.php/ap http://ojs.cvut.cz/ojs/index.php/ap PROPN ap-2081 1 23 on on ADP ap-2081 1 24 the the DET ap-2081 1 25 real real ADJ ap-2081 1 26 matrix matrix NOUN ap-2081 1 27 representation representation NOUN ap-2081 1 28 of of ADP ap-2081 1 29 pt pt ADJ ap-2081 1 30 - - ADJ ap-2081 1 31 symmetric symmetric ADJ ap-2081 1 32 operators operator NOUN ap-2081 1 33 francisco francisco PROPN ap-2081 1 34 m. m. PROPN ap-2081 1 35 fernández fernández PROPN ap-2081 1 36 inifta inifta PROPN ap-2081 1 37 ( ( PUNCT ap-2081 1 38 unlp unlp ADJ ap-2081 1 39 , , PUNCT ap-2081 1 40 cct cct PROPN ap-2081 1 41 la la PROPN ap-2081 1 42 plata plata PROPN ap-2081 1 43 - - PUNCT ap-2081 1 44 conicet conicet NOUN ap-2081 1 45 ) ) PUNCT ap-2081 1 46 , , PUNCT ap-2081 1 47 blvd blvd NOUN ap-2081 1 48 . . PUNCT ap-2081 2 1 113 113 NUM ap-2081 2 2 y y NUM ap-2081 2 3 64 64 NUM ap-2081 2 4 s s NOUN ap-2081 2 5 / / SYM ap-2081 2 6 n n CCONJ ap-2081 2 7 , , PUNCT ap-2081 2 8 sucursal sucursal ADJ ap-2081 2 9 4 4 NUM ap-2081 2 10 , , PUNCT ap-2081 2 11 casilla casilla X ap-2081 2 12 de de X ap-2081 2 13 correo correo PROPN ap-2081 2 14 16 16 NUM ap-2081 2 15 , , PUNCT ap-2081 2 16 1900 1900 NUM ap-2081 2 17 la la PROPN ap-2081 2 18 plata plata PROPN ap-2081 2 19 , , PUNCT ap-2081 2 20 argentina argentina PROPN ap-2081 2 21 correspondence correspondence NOUN ap-2081 2 22 : : PUNCT ap-2081 2 23 fernande@quimica.unlp.edu.ar fernande@quimica.unlp.edu.ar VERB ap-2081 2 24 abstract abstract ADJ ap-2081 2 25 . . PUNCT ap-2081 3 1 we we PRON ap-2081 3 2 discuss discuss VERB ap-2081 3 3 the the DET ap-2081 3 4 construction construction NOUN ap-2081 3 5 of of ADP ap-2081 3 6 real real ADJ ap-2081 3 7 matrix matrix NOUN ap-2081 3 8 representations representation NOUN ap-2081 3 9 of of ADP ap-2081 3 10 pt pt ADJ ap-2081 3 11 - - PUNCT ap-2081 3 12 symmetric symmetric ADJ ap-2081 3 13 operators operator NOUN ap-2081 3 14 . . PUNCT ap-2081 4 1 we we PRON ap-2081 4 2 show show VERB ap-2081 4 3 the the DET ap-2081 4 4 limitation limitation NOUN ap-2081 4 5 of of ADP ap-2081 4 6 a a DET ap-2081 4 7 general general ADJ ap-2081 4 8 recipe recipe NOUN ap-2081 4 9 presented present VERB ap-2081 4 10 some some DET ap-2081 4 11 time time NOUN ap-2081 4 12 ago ago ADV ap-2081 4 13 for for ADP ap-2081 4 14 non non ADJ ap-2081 4 15 - - ADJ ap-2081 4 16 hermitian hermitian ADJ ap-2081 4 17 hamiltonians hamiltonian NOUN ap-2081 4 18 with with ADP ap-2081 4 19 antiunitary antiunitary ADJ ap-2081 4 20 symmetry symmetry NOUN ap-2081 4 21 and and CCONJ ap-2081 4 22 propose propose VERB ap-2081 4 23 a a DET ap-2081 4 24 way way NOUN ap-2081 4 25 to to PART ap-2081 4 26 overcome overcome VERB ap-2081 4 27 it it PRON ap-2081 4 28 . . PUNCT ap-2081 5 1 our our PRON ap-2081 5 2 results result NOUN ap-2081 5 3 agree agree VERB ap-2081 5 4 with with ADP ap-2081 5 5 earlier early ADJ ap-2081 5 6 ones one NOUN ap-2081 5 7 for for ADP ap-2081 5 8 a a DET ap-2081 5 9 particular particular ADJ ap-2081 5 10 case case NOUN ap-2081 5 11 . . PUNCT ap-2081 6 1 keywords keyword NOUN ap-2081 6 2 : : PUNCT ap-2081 6 3 non non ADJ ap-2081 6 4 - - ADJ ap-2081 6 5 hermitian hermitian ADJ ap-2081 6 6 hamiltonians hamiltonian NOUN ap-2081 6 7 , , PUNCT ap-2081 6 8 antiunitary antiunitary NOUN ap-2081 6 9 symmetry symmetry NOUN ap-2081 6 10 , , PUNCT ap-2081 6 11 pt pt NOUN ap-2081 6 12 - - PUNCT ap-2081 6 13 symmetry symmetry NOUN ap-2081 6 14 , , PUNCT ap-2081 6 15 real real ADJ ap-2081 6 16 matrix matrix NOUN ap-2081 6 17 . . PUNCT ap-2081 7 1 1 1 X ap-2081 7 2 . . X ap-2081 7 3 introduction introduction NOUN ap-2081 7 4 at at ADP ap-2081 7 5 first first ADJ ap-2081 7 6 sight sight NOUN ap-2081 7 7 it it PRON ap-2081 7 8 is be AUX ap-2081 7 9 suprising suprise VERB ap-2081 7 10 that that SCONJ ap-2081 7 11 a a DET ap-2081 7 12 subset subset NOUN ap-2081 7 13 of of ADP ap-2081 7 14 eigenvalues eigenvalue NOUN ap-2081 7 15 of of ADP ap-2081 7 16 a a DET ap-2081 7 17 complex complex ADV ap-2081 7 18 - - PUNCT ap-2081 7 19 valued value VERB ap-2081 7 20 non non ADJ ap-2081 7 21 - - ADJ ap-2081 7 22 hermitian hermitian ADJ ap-2081 7 23 operator operator NOUN ap-2081 7 24 ĥ ĥ PUNCT ap-2081 7 25 can can AUX ap-2081 7 26 be be AUX ap-2081 7 27 real real ADJ ap-2081 7 28 ( ( PUNCT ap-2081 7 29 see see VERB ap-2081 7 30 [ [ X ap-2081 7 31 1 1 X ap-2081 7 32 ] ] PUNCT ap-2081 7 33 and and CCONJ ap-2081 7 34 references reference NOUN ap-2081 7 35 therein therein ADV ap-2081 7 36 ) ) PUNCT ap-2081 7 37 . . PUNCT ap-2081 8 1 in in ADP ap-2081 8 2 order order NOUN ap-2081 8 3 to to PART ap-2081 8 4 provide provide VERB ap-2081 8 5 a a DET ap-2081 8 6 simple simple ADJ ap-2081 8 7 and and CCONJ ap-2081 8 8 general general ADJ ap-2081 8 9 explanation explanation NOUN ap-2081 8 10 of of ADP ap-2081 8 11 this this DET ap-2081 8 12 fact fact NOUN ap-2081 8 13 bender bender NOUN ap-2081 8 14 et et PROPN ap-2081 8 15 al al PROPN ap-2081 8 16 . . PUNCT ap-2081 9 1 [ [ X ap-2081 9 2 2 2 X ap-2081 9 3 ] ] PUNCT ap-2081 9 4 showed show VERB ap-2081 9 5 that that SCONJ ap-2081 9 6 it it PRON ap-2081 9 7 is be AUX ap-2081 9 8 possible possible ADJ ap-2081 9 9 to to PART ap-2081 9 10 construct construct VERB ap-2081 9 11 a a DET ap-2081 9 12 basis basis NOUN ap-2081 9 13 set set NOUN ap-2081 9 14 of of ADP ap-2081 9 15 vectors vector NOUN ap-2081 9 16 so so SCONJ ap-2081 9 17 that that SCONJ ap-2081 9 18 the the DET ap-2081 9 19 matrix matrix NOUN ap-2081 9 20 representation representation NOUN ap-2081 9 21 of of ADP ap-2081 9 22 such such DET ap-2081 9 23 an an DET ap-2081 9 24 operator operator NOUN ap-2081 9 25 is be AUX ap-2081 9 26 real real ADJ ap-2081 9 27 . . PUNCT ap-2081 10 1 as as ADP ap-2081 10 2 a a DET ap-2081 10 3 result result NOUN ap-2081 10 4 the the DET ap-2081 10 5 secular secular ADJ ap-2081 10 6 determinant determinant ADJ ap-2081 10 7 is be AUX ap-2081 10 8 real real ADJ ap-2081 10 9 ( ( PUNCT ap-2081 10 10 the the DET ap-2081 10 11 coefficients coefficient NOUN ap-2081 10 12 of of ADP ap-2081 10 13 the the DET ap-2081 10 14 characteristic characteristic ADJ ap-2081 10 15 polynomial polynomial NOUN ap-2081 10 16 are be AUX ap-2081 10 17 real real ADJ ap-2081 10 18 ) ) PUNCT ap-2081 10 19 and and CCONJ ap-2081 10 20 its its PRON ap-2081 10 21 roots root NOUN ap-2081 10 22 are be AUX ap-2081 10 23 either either CCONJ ap-2081 10 24 real real ADJ ap-2081 10 25 or or CCONJ ap-2081 10 26 appear appear VERB ap-2081 10 27 in in ADP ap-2081 10 28 pairs pair NOUN ap-2081 10 29 of of ADP ap-2081 10 30 complex complex ADJ ap-2081 10 31 conjugate conjugate ADJ ap-2081 10 32 numbers number NOUN ap-2081 10 33 . . PUNCT ap-2081 11 1 the the DET ap-2081 11 2 argument argument NOUN ap-2081 11 3 is be AUX ap-2081 11 4 based base VERB ap-2081 11 5 on on ADP ap-2081 11 6 the the DET ap-2081 11 7 existence existence NOUN ap-2081 11 8 of of ADP ap-2081 11 9 an an DET ap-2081 11 10 antiunitary antiunitary ADJ ap-2081 11 11 symmetry symmetry NOUN ap-2081 11 12 âĥâ−1 âĥâ−1 NOUN ap-2081 11 13 = = SYM ap-2081 11 14 ĥ ĥ PROPN ap-2081 11 15 , , PUNCT ap-2081 11 16 where where SCONJ ap-2081 11 17 the the DET ap-2081 11 18 antiunitary antiunitary ADJ ap-2081 11 19 operator operator NOUN ap-2081 11 20 â â ADP ap-2081 11 21 satisfies satisfy VERB ap-2081 11 22 âk âk X ap-2081 11 23 = = SYM ap-2081 11 24 1̂ 1̂ PROPN ap-2081 11 25 for for ADP ap-2081 11 26 k k PROPN ap-2081 11 27 odd odd ADJ ap-2081 11 28 . . PUNCT ap-2081 12 1 bender bender PROPN ap-2081 12 2 et et PROPN ap-2081 12 3 al al PROPN ap-2081 12 4 . . PUNCT ap-2081 13 1 [ [ X ap-2081 13 2 2 2 NUM ap-2081 13 3 ] ] PUNCT ap-2081 13 4 showed show VERB ap-2081 13 5 some some DET ap-2081 13 6 illustrative illustrative ADJ ap-2081 13 7 examples example NOUN ap-2081 13 8 of of ADP ap-2081 13 9 their their PRON ap-2081 13 10 general general ADJ ap-2081 13 11 result result NOUN ap-2081 13 12 . . PUNCT ap-2081 14 1 the the DET ap-2081 14 2 procedure procedure NOUN ap-2081 14 3 followed follow VERB ap-2081 14 4 by by ADP ap-2081 14 5 bender bender PROPN ap-2081 14 6 et et PROPN ap-2081 14 7 al al PROPN ap-2081 14 8 . . PUNCT ap-2081 15 1 [ [ X ap-2081 15 2 2 2 X ap-2081 15 3 ] ] PUNCT ap-2081 15 4 for for ADP ap-2081 15 5 the the DET ap-2081 15 6 construction construction NOUN ap-2081 15 7 of of ADP ap-2081 15 8 the the DET ap-2081 15 9 suitable suitable ADJ ap-2081 15 10 basis basis NOUN ap-2081 15 11 set set NOUN ap-2081 15 12 is be AUX ap-2081 15 13 reminiscent reminiscent ADJ ap-2081 15 14 of of ADP ap-2081 15 15 the the DET ap-2081 15 16 one one NOUN ap-2081 15 17 used use VERB ap-2081 15 18 by by ADP ap-2081 15 19 porter porter NOUN ap-2081 15 20 [ [ X ap-2081 15 21 3 3 X ap-2081 15 22 ] ] PUNCT ap-2081 15 23 in in ADP ap-2081 15 24 the the DET ap-2081 15 25 study study NOUN ap-2081 15 26 of of ADP ap-2081 15 27 matrix matrix NOUN ap-2081 15 28 representations representation NOUN ap-2081 15 29 of of ADP ap-2081 15 30 hermitian hermitian ADJ ap-2081 15 31 operators operator NOUN ap-2081 15 32 . . PUNCT ap-2081 16 1 however however ADV ap-2081 16 2 , , PUNCT ap-2081 16 3 the the DET ap-2081 16 4 ansatz ansatz NOUN ap-2081 16 5 proposed propose VERB ap-2081 16 6 by by ADP ap-2081 16 7 porter porter NOUN ap-2081 16 8 appears appear VERB ap-2081 16 9 to to PART ap-2081 16 10 be be AUX ap-2081 16 11 somewhat somewhat ADV ap-2081 16 12 more more ADV ap-2081 16 13 general general ADJ ap-2081 16 14 . . PUNCT ap-2081 17 1 the the DET ap-2081 17 2 purpose purpose NOUN ap-2081 17 3 of of ADP ap-2081 17 4 this this DET ap-2081 17 5 paper paper NOUN ap-2081 17 6 is be AUX ap-2081 17 7 to to PART ap-2081 17 8 analyse analyse VERB ap-2081 17 9 the the DET ap-2081 17 10 argument argument NOUN ap-2081 17 11 given give VERB ap-2081 17 12 by by ADP ap-2081 17 13 bender bender PROPN ap-2081 17 14 et et PROPN ap-2081 17 15 al al PROPN ap-2081 17 16 . . PUNCT ap-2081 18 1 [ [ X ap-2081 18 2 2 2 X ap-2081 18 3 ] ] PUNCT ap-2081 18 4 in in ADP ap-2081 18 5 more more ADJ ap-2081 18 6 detail detail NOUN ap-2081 18 7 . . PUNCT ap-2081 19 1 in in ADP ap-2081 19 2 section section NOUN ap-2081 19 3 2 2 NUM ap-2081 19 4 we we PRON ap-2081 19 5 outline outline VERB ap-2081 19 6 the the DET ap-2081 19 7 main main ADJ ap-2081 19 8 features feature NOUN ap-2081 19 9 of of ADP ap-2081 19 10 an an DET ap-2081 19 11 antiunitary antiunitary NOUN ap-2081 19 12 or or CCONJ ap-2081 19 13 antilinear antilinear ADJ ap-2081 19 14 operator operator NOUN ap-2081 19 15 and and CCONJ ap-2081 19 16 in in ADP ap-2081 19 17 section section NOUN ap-2081 19 18 3 3 NUM ap-2081 19 19 we we PRON ap-2081 19 20 briefly briefly ADV ap-2081 19 21 discuss discuss VERB ap-2081 19 22 the the DET ap-2081 19 23 concept concept NOUN ap-2081 19 24 of of ADP ap-2081 19 25 antiunitary antiunitary ADJ ap-2081 19 26 symmetry symmetry NOUN ap-2081 19 27 . . PUNCT ap-2081 20 1 in in ADP ap-2081 20 2 section section NOUN ap-2081 20 3 4 4 NUM ap-2081 20 4 we we PRON ap-2081 20 5 review review VERB ap-2081 20 6 the the DET ap-2081 20 7 argument argument NOUN ap-2081 20 8 given give VERB ap-2081 20 9 by by ADP ap-2081 20 10 bender bender PROPN ap-2081 20 11 et et PROPN ap-2081 20 12 al al PROPN ap-2081 20 13 . . PUNCT ap-2081 21 1 [ [ X ap-2081 21 2 2 2 X ap-2081 21 3 ] ] PUNCT ap-2081 21 4 and and CCONJ ap-2081 21 5 show show VERB ap-2081 21 6 that that SCONJ ap-2081 21 7 under under ADP ap-2081 21 8 certain certain ADJ ap-2081 21 9 conditions condition NOUN ap-2081 21 10 it it PRON ap-2081 21 11 does do AUX ap-2081 21 12 not not PART ap-2081 21 13 apply apply VERB ap-2081 21 14 . . PUNCT ap-2081 22 1 we we PRON ap-2081 22 2 illustrate illustrate VERB ap-2081 22 3 this this DET ap-2081 22 4 point point NOUN ap-2081 22 5 by by ADP ap-2081 22 6 means mean NOUN ap-2081 22 7 of of ADP ap-2081 22 8 the the DET ap-2081 22 9 well well ADV ap-2081 22 10 known know VERB ap-2081 22 11 harmonic harmonic ADJ ap-2081 22 12 - - PUNCT ap-2081 22 13 oscillator oscillator NOUN ap-2081 22 14 basis basis NOUN ap-2081 22 15 set set NOUN ap-2081 22 16 and and CCONJ ap-2081 22 17 show show VERB ap-2081 22 18 how how SCONJ ap-2081 22 19 to to PART ap-2081 22 20 overcome overcome VERB ap-2081 22 21 that that DET ap-2081 22 22 shortcoming shortcoming NOUN ap-2081 22 23 . . PUNCT ap-2081 23 1 in in ADP ap-2081 23 2 section section NOUN ap-2081 23 3 5 5 NUM ap-2081 23 4 we we PRON ap-2081 23 5 discuss discuss VERB ap-2081 23 6 the the DET ap-2081 23 7 harmonic harmonic ADJ ap-2081 23 8 - - PUNCT ap-2081 23 9 oscillator oscillator NOUN ap-2081 23 10 basis basis NOUN ap-2081 23 11 set set VERB ap-2081 23 12 in in ADP ap-2081 23 13 more more ADJ ap-2081 23 14 detail detail NOUN ap-2081 23 15 and and CCONJ ap-2081 23 16 in in ADP ap-2081 23 17 section section NOUN ap-2081 23 18 6 6 NUM ap-2081 23 19 we we PRON ap-2081 23 20 draw draw VERB ap-2081 23 21 conclusions conclusion NOUN ap-2081 23 22 . . PUNCT ap-2081 24 1 2 2 X ap-2081 24 2 . . X ap-2081 24 3 antiunitary antiunitary ADJ ap-2081 24 4 operator operator NOUN ap-2081 24 5 as as SCONJ ap-2081 24 6 already already ADV ap-2081 24 7 mentioned mention VERB ap-2081 24 8 above above ADV ap-2081 24 9 , , PUNCT ap-2081 24 10 a a DET ap-2081 24 11 wide wide ADJ ap-2081 24 12 class class NOUN ap-2081 24 13 of of ADP ap-2081 24 14 nonhermitian nonhermitian ADJ ap-2081 24 15 hamiltonians hamiltonian NOUN ap-2081 24 16 with with ADP ap-2081 24 17 unbroken unbroken ADJ ap-2081 24 18 pt pt NOUN ap-2081 24 19 symmetry symmetry NOUN ap-2081 24 20 exhibits exhibit VERB ap-2081 24 21 real real ADJ ap-2081 24 22 spectra spectra NOUN ap-2081 24 23 [ [ X ap-2081 24 24 1 1 NUM ap-2081 24 25 ] ] PUNCT ap-2081 24 26 . . PUNCT ap-2081 25 1 in in ADP ap-2081 25 2 general general ADJ ap-2081 25 3 , , PUNCT ap-2081 25 4 they they PRON ap-2081 25 5 are be AUX ap-2081 25 6 invariant invariant ADJ ap-2081 25 7 under under ADP ap-2081 25 8 an an DET ap-2081 25 9 antilinear antilinear ADJ ap-2081 25 10 or or CCONJ ap-2081 25 11 antiunitary antiunitary ADJ ap-2081 25 12 transformation transformation NOUN ap-2081 25 13 of of ADP ap-2081 25 14 the the DET ap-2081 25 15 form form NOUN ap-2081 25 16 â−1ĥâ â−1ĥâ NOUN ap-2081 25 17 = = SYM ap-2081 25 18 ĥ. ĥ. NOUN ap-2081 25 19 the the DET ap-2081 25 20 antiunitary antiunitary ADJ ap-2081 25 21 operator operator NOUN ap-2081 25 22 â â ADP ap-2081 25 23 satisfies satisfie NOUN ap-2081 25 24 [ [ X ap-2081 25 25 4 4 NUM ap-2081 25 26 ] ] PUNCT ap-2081 25 27 â â X ap-2081 25 28 ( ( PUNCT ap-2081 25 29 |f〉+ |f〉+ PROPN ap-2081 25 30 |g |g PROPN ap-2081 25 31 〉 〉 PROPN ap-2081 25 32 ) ) PUNCT ap-2081 26 1 = = SYM ap-2081 26 2 â|f〉+ â|f〉+ PROPN ap-2081 26 3 â|g â|g NOUN ap-2081 26 4 〉 〉 NOUN ap-2081 26 5 âc|f âc|f ADV ap-2081 26 6 〉 〉 NOUN ap-2081 26 7 = = SYM ap-2081 26 8 c∗â|f c∗â|f PROPN ap-2081 26 9 〉 〉 NOUN ap-2081 26 10 , , PUNCT ap-2081 26 11 ( ( PUNCT ap-2081 26 12 1 1 X ap-2081 26 13 ) ) PUNCT ap-2081 26 14 for for ADP ap-2081 26 15 any any DET ap-2081 26 16 pair pair NOUN ap-2081 26 17 of of ADP ap-2081 26 18 vectors vector NOUN ap-2081 26 19 ∣∣f ∣∣f NOUN ap-2081 26 20 〉 〉 NOUN ap-2081 26 21 and and CCONJ ap-2081 26 22 ∣∣g ∣∣g PROPN ap-2081 26 23 〉 〉 NOUN ap-2081 26 24 and and CCONJ ap-2081 26 25 arbitrary arbitrary ADJ ap-2081 26 26 complex complex ADJ ap-2081 26 27 number number NOUN ap-2081 26 28 c c NOUN ap-2081 26 29 , , PUNCT ap-2081 26 30 where where SCONJ ap-2081 26 31 the the DET ap-2081 26 32 asterisk asterisk NOUN ap-2081 26 33 denotes denote VERB ap-2081 26 34 complex complex ADJ ap-2081 26 35 conjugation conjugation NOUN ap-2081 26 36 . . PUNCT ap-2081 27 1 this this DET ap-2081 27 2 definition definition NOUN ap-2081 27 3 is be AUX ap-2081 27 4 equivalent equivalent ADJ ap-2081 27 5 to to ADP ap-2081 27 6 〈 〈 PROPN ap-2081 27 7 âf âf NUM ap-2081 27 8 ∣∣âg ∣∣âg PROPN ap-2081 27 9 〉 〉 NOUN ap-2081 27 10 = = SYM ap-2081 27 11 〈 〈 PROPN ap-2081 27 12 f f X ap-2081 27 13 |g〉∗. |g〉∗. PROPN ap-2081 27 14 ( ( PUNCT ap-2081 27 15 2 2 X ap-2081 27 16 ) ) PUNCT ap-2081 27 17 one one NOUN ap-2081 27 18 can can AUX ap-2081 27 19 easily easily ADV ap-2081 27 20 derive derive VERB ap-2081 27 21 the the DET ap-2081 27 22 pair pair NOUN ap-2081 27 23 of of ADP ap-2081 27 24 equations equation NOUN ap-2081 27 25 ( ( PUNCT ap-2081 27 26 1 1 NUM ap-2081 27 27 ) ) PUNCT ap-2081 27 28 from from ADP ap-2081 27 29 ( ( PUNCT ap-2081 27 30 2 2 NUM ap-2081 27 31 ) ) PUNCT ap-2081 27 32 so so SCONJ ap-2081 27 33 that that SCONJ ap-2081 27 34 the the DET ap-2081 27 35 latter latter NOUN ap-2081 27 36 can can AUX ap-2081 27 37 be be AUX ap-2081 27 38 considered consider VERB ap-2081 27 39 to to PART ap-2081 27 40 be be AUX ap-2081 27 41 the the DET ap-2081 27 42 actual actual ADJ ap-2081 27 43 definition definition NOUN ap-2081 27 44 of of ADP ap-2081 27 45 an an DET ap-2081 27 46 antiunitary antiunitary ADJ ap-2081 27 47 operator operator NOUN ap-2081 27 48 [ [ X ap-2081 27 49 4 4 NUM ap-2081 27 50 ] ] PUNCT ap-2081 27 51 . . PUNCT ap-2081 28 1 if if SCONJ ap-2081 28 2 k̂ k̂ PROPN ap-2081 28 3 is be AUX ap-2081 28 4 an an DET ap-2081 28 5 antilinear antilinear ADJ ap-2081 28 6 operator operator NOUN ap-2081 28 7 such such ADJ ap-2081 28 8 that that SCONJ ap-2081 28 9 k̂2 k̂2 PROPN ap-2081 29 1 = = SYM ap-2081 29 2 1̂ 1̂ NOUN ap-2081 29 3 ( ( PUNCT ap-2081 29 4 for for ADP ap-2081 29 5 example example NOUN ap-2081 29 6 , , PUNCT ap-2081 29 7 the the DET ap-2081 29 8 complex complex ADJ ap-2081 29 9 conjugation conjugation NOUN ap-2081 29 10 operator operator NOUN ap-2081 29 11 ) ) PUNCT ap-2081 29 12 then then ADV ap-2081 29 13 it it PRON ap-2081 29 14 follows follow VERB ap-2081 29 15 from from ADP ap-2081 29 16 ( ( PUNCT ap-2081 29 17 2 2 NUM ap-2081 29 18 ) ) PUNCT ap-2081 29 19 that that PRON ap-2081 29 20 âk̂ âk̂ VERB ap-2081 30 1 = = NOUN ap-2081 30 2 û û X ap-2081 30 3 is be AUX ap-2081 30 4 unitary unitary ADJ ap-2081 30 5 ( ( PUNCT ap-2081 30 6 û† û† SYM ap-2081 30 7 = = SYM ap-2081 30 8 û−1 û−1 PROPN ap-2081 30 9 ) ) PUNCT ap-2081 30 10 ; ; PUNCT ap-2081 30 11 that that PRON ap-2081 30 12 is be AUX ap-2081 30 13 to to PART ap-2081 30 14 say say VERB ap-2081 30 15 the the DET ap-2081 30 16 inner inner ADJ ap-2081 30 17 product product NOUN ap-2081 30 18 〈 〈 PROPN ap-2081 30 19 f f PROPN ap-2081 30 20 ∣∣g ∣∣g PROPN ap-2081 30 21 〉 〉 NOUN ap-2081 30 22 remains remain VERB ap-2081 30 23 invariant invariant ADJ ap-2081 30 24 under under ADP ap-2081 30 25 û û NUM ap-2081 30 26 : : PUNCT ap-2081 30 27 〈 〈 PROPN ap-2081 30 28 âk̂f âk̂f ADV ap-2081 30 29 ∣∣âk̂g ∣∣âk̂g ADJ ap-2081 30 30 〉 〉 NOUN ap-2081 30 31 = = SYM ap-2081 30 32 〈 〈 PROPN ap-2081 30 33 k̂f k̂f NOUN ap-2081 30 34 ∣∣k̂g〉∗ ∣∣k̂g〉∗ NOUN ap-2081 30 35 = = SYM ap-2081 30 36 〈 〈 PROPN ap-2081 30 37 f f PROPN ap-2081 30 38 |g |g PROPN ap-2081 30 39 〉 〉 PROPN ap-2081 30 40 . . PUNCT ap-2081 31 1 ( ( PUNCT ap-2081 31 2 3 3 X ap-2081 31 3 ) ) PUNCT ap-2081 31 4 in in ADP ap-2081 31 5 other other ADJ ap-2081 31 6 words word NOUN ap-2081 31 7 , , PUNCT ap-2081 31 8 any any DET ap-2081 31 9 antilinear antilinear ADJ ap-2081 31 10 operator operator NOUN ap-2081 31 11 â â PRON ap-2081 31 12 can can AUX ap-2081 31 13 be be AUX ap-2081 31 14 written write VERB ap-2081 31 15 as as ADP ap-2081 31 16 a a DET ap-2081 31 17 product product NOUN ap-2081 31 18 of of ADP ap-2081 31 19 a a DET ap-2081 31 20 unitary unitary ADJ ap-2081 31 21 operator operator NOUN ap-2081 31 22 and and CCONJ ap-2081 31 23 the the DET ap-2081 31 24 complex complex ADJ ap-2081 31 25 conjugation conjugation NOUN ap-2081 31 26 operation operation NOUN ap-2081 32 1 [ [ X ap-2081 32 2 4 4 NUM ap-2081 32 3 ] ] PUNCT ap-2081 32 4 . . PUNCT ap-2081 33 1 in in ADP ap-2081 33 2 exactly exactly ADV ap-2081 33 3 in in ADP ap-2081 33 4 the the DET ap-2081 33 5 same same ADJ ap-2081 33 6 way way NOUN ap-2081 33 7 we we PRON ap-2081 33 8 can can AUX ap-2081 33 9 easily easily ADV ap-2081 33 10 prove prove VERB ap-2081 33 11 that that SCONJ ap-2081 33 12 â2j â2j ADJ ap-2081 33 13 is be AUX ap-2081 33 14 unitary unitary ADJ ap-2081 33 15 and and CCONJ ap-2081 33 16 â2j+1 â2j+1 ADJ ap-2081 33 17 antiunitary antiunitary NOUN ap-2081 33 18 . . PUNCT ap-2081 34 1 in in ADP ap-2081 34 2 their their PRON ap-2081 34 3 discussion discussion NOUN ap-2081 34 4 of of ADP ap-2081 34 5 real real ADJ ap-2081 34 6 matrix matrix NOUN ap-2081 34 7 representations representation NOUN ap-2081 34 8 of of ADP ap-2081 34 9 non non ADJ ap-2081 34 10 - - ADJ ap-2081 34 11 hermitian hermitian ADJ ap-2081 34 12 hamiltonians hamiltonian NOUN ap-2081 34 13 bender bender PROPN ap-2081 34 14 et et PROPN ap-2081 34 15 al al PROPN ap-2081 34 16 . . PUNCT ap-2081 35 1 [ [ X ap-2081 35 2 2 2 X ap-2081 35 3 ] ] PUNCT ap-2081 35 4 considered consider VERB ap-2081 35 5 hamiltonians hamiltonian NOUN ap-2081 35 6 ĥ ĥ PUNCT ap-2081 35 7 with with ADP ap-2081 35 8 antiunitary antiunitary ADJ ap-2081 35 9 symmetry symmetry NOUN ap-2081 35 10 âĥâ−1 âĥâ−1 NOUN ap-2081 35 11 = = SYM ap-2081 35 12 ĥ ĥ X ap-2081 35 13 , , PUNCT ap-2081 35 14 ( ( PUNCT ap-2081 35 15 4 4 NUM ap-2081 35 16 ) ) PUNCT ap-2081 35 17 where where SCONJ ap-2081 35 18 â â PRON ap-2081 35 19 satisfies satisfy VERB ap-2081 35 20 the the DET ap-2081 35 21 additional additional ADJ ap-2081 35 22 condition condition NOUN ap-2081 35 23 â2k â2k ADP ap-2081 35 24 = = SYM ap-2081 35 25 1̂ 1̂ PROPN ap-2081 35 26 , , PUNCT ap-2081 35 27 k k PROPN ap-2081 35 28 odd odd ADJ ap-2081 35 29 . . PUNCT ap-2081 36 1 ( ( PUNCT ap-2081 36 2 5 5 NUM ap-2081 36 3 ) ) PUNCT ap-2081 36 4 since since SCONJ ap-2081 36 5 b̂ b̂ NOUN ap-2081 36 6 = = PRON ap-2081 36 7 âk âk X ap-2081 36 8 is be AUX ap-2081 36 9 antiunitary antiunitary ADJ ap-2081 36 10 and and CCONJ ap-2081 36 11 satisfies satisfy VERB ap-2081 36 12 b̂2 b̂2 X ap-2081 37 1 = = PUNCT ap-2081 37 2 1̂ 1̂ NOUN ap-2081 37 3 we we PRON ap-2081 37 4 can can AUX ap-2081 37 5 restrict restrict VERB ap-2081 37 6 our our PRON ap-2081 37 7 discussion discussion NOUN ap-2081 37 8 to to ADP ap-2081 37 9 the the DET ap-2081 37 10 case case NOUN ap-2081 37 11 k k NOUN ap-2081 38 1 = = SYM ap-2081 38 2 1 1 NUM ap-2081 38 3 without without ADP ap-2081 38 4 loss loss NOUN ap-2081 38 5 of of ADP ap-2081 38 6 generality generality NOUN ap-2081 38 7 . . PUNCT ap-2081 39 1 therefore therefore ADV ap-2081 39 2 , , PUNCT ap-2081 39 3 from from ADP ap-2081 39 4 now now ADV ap-2081 39 5 on on ADV ap-2081 39 6 we we PRON ap-2081 39 7 substitute substitute VERB ap-2081 39 8 the the DET ap-2081 39 9 condition condition NOUN ap-2081 39 10 â2 â2 PUNCT ap-2081 40 1 = = SYM ap-2081 40 2 1̂ 1̂ NUM ap-2081 40 3 ( ( PUNCT ap-2081 40 4 6 6 NUM ap-2081 40 5 ) ) SYM ap-2081 40 6 113 113 NUM ap-2081 40 7 http://dx.doi.org/10.14311/ap.2014.54.0113 http://dx.doi.org/10.14311/ap.2014.54.0113 NOUN ap-2081 40 8 http://ojs.cvut.cz/ojs/index.php/ap http://ojs.cvut.cz/ojs/index.php/ap PROPN ap-2081 40 9 francisco francisco PROPN ap-2081 40 10 m. m. NOUN ap-2081 40 11 fernández fernández PROPN ap-2081 40 12 acta acta PROPN ap-2081 40 13 polytechnica polytechnica PROPN ap-2081 40 14 for for ADP ap-2081 40 15 the the DET ap-2081 40 16 apparently apparently ADV ap-2081 40 17 more more ADV ap-2081 40 18 general general ADJ ap-2081 40 19 equation equation NOUN ap-2081 40 20 ( ( PUNCT ap-2081 40 21 5 5 NUM ap-2081 40 22 ) ) PUNCT ap-2081 40 23 . . PUNCT ap-2081 41 1 from from ADP ap-2081 41 2 now now ADV ap-2081 41 3 on on ADV ap-2081 41 4 we we PRON ap-2081 41 5 refer refer VERB ap-2081 41 6 to to ADP ap-2081 41 7 equation equation NOUN ap-2081 41 8 ( ( PUNCT ap-2081 41 9 4 4 NUM ap-2081 41 10 ) ) PUNCT ap-2081 41 11 as as ADP ap-2081 41 12 a a DET ap-2081 41 13 - - PUNCT ap-2081 41 14 symmetry symmetry NOUN ap-2081 41 15 and and CCONJ ap-2081 41 16 to to ADP ap-2081 41 17 the the DET ap-2081 41 18 operator operator NOUN ap-2081 41 19 ĥ ĥ PUNCT ap-2081 41 20 as as ADP ap-2081 41 21 a a DET ap-2081 41 22 - - PUNCT ap-2081 41 23 symmetric symmetric NOUN ap-2081 41 24 for for ADP ap-2081 41 25 short short ADJ ap-2081 41 26 . . PUNCT ap-2081 42 1 3 3 X ap-2081 42 2 . . X ap-2081 42 3 antiunitary antiunitary ADJ ap-2081 42 4 symmetry symmetry NOUN ap-2081 42 5 it it PRON ap-2081 42 6 follows follow VERB ap-2081 42 7 from from ADP ap-2081 42 8 the the DET ap-2081 42 9 antiunitary antiunitary ADJ ap-2081 42 10 invariance invariance NOUN ap-2081 42 11 ( ( PUNCT ap-2081 42 12 4 4 NUM ap-2081 42 13 ) ) PUNCT ap-2081 43 1 that that SCONJ ap-2081 43 2 [ [ X ap-2081 43 3 ĥ ĥ NOUN ap-2081 43 4 , , PUNCT ap-2081 43 5 â â ADP ap-2081 43 6 ] ] X ap-2081 43 7 = = SYM ap-2081 43 8 0 0 X ap-2081 43 9 . . PUNCT ap-2081 44 1 therefore therefore ADV ap-2081 44 2 , , PUNCT ap-2081 44 3 if if SCONJ ap-2081 44 4 |ψ |ψ PROPN ap-2081 44 5 〉 〉 PROPN ap-2081 44 6 is be AUX ap-2081 44 7 an an DET ap-2081 44 8 eigenvector eigenvector NOUN ap-2081 44 9 of of ADP ap-2081 44 10 ĥ ĥ PUNCT ap-2081 44 11 with with ADP ap-2081 44 12 eigenvalue eigenvalue PROPN ap-2081 44 13 e e NOUN ap-2081 44 14 ĥ|ψ ĥ|ψ VERB ap-2081 44 15 〉 〉 NOUN ap-2081 44 16 = = PUNCT ap-2081 44 17 e|ψ e|ψ PROPN ap-2081 44 18 〉 〉 PROPN ap-2081 44 19 , , PUNCT ap-2081 44 20 ( ( PUNCT ap-2081 44 21 7 7 X ap-2081 44 22 ) ) PUNCT ap-2081 44 23 we we PRON ap-2081 44 24 have have VERB ap-2081 44 25 [ [ X ap-2081 44 26 ĥ ĥ X ap-2081 44 27 , , PUNCT ap-2081 44 28 â]|ψ â]|ψ NOUN ap-2081 44 29 〉 〉 NOUN ap-2081 44 30 = = PUNCT ap-2081 44 31 ĥâ|ψ ĥâ|ψ PROPN ap-2081 44 32 〉 〉 PROPN ap-2081 44 33 − − NOUN ap-2081 44 34 âĥ|ψ âĥ|ψ PROPN ap-2081 44 35 〉 〉 NOUN ap-2081 44 36 = = PUNCT ap-2081 44 37 ĥâ|ψ ĥâ|ψ PROPN ap-2081 44 38 〉 〉 NOUN ap-2081 44 39 − − PROPN ap-2081 44 40 e∗â|ψ e∗â|ψ NOUN ap-2081 44 41 〉 〉 NOUN ap-2081 44 42 = = SYM ap-2081 44 43 0 0 PROPN ap-2081 44 44 . . PUNCT ap-2081 45 1 ( ( PUNCT ap-2081 45 2 8) 8) NUM ap-2081 45 3 this this DET ap-2081 45 4 equation equation NOUN ap-2081 45 5 tells tell VERB ap-2081 45 6 us we PRON ap-2081 45 7 that that SCONJ ap-2081 45 8 if if SCONJ ap-2081 45 9 ∣∣ψ ∣∣ψ NOUN ap-2081 45 10 〉 〉 NOUN ap-2081 45 11 is be AUX ap-2081 45 12 an an DET ap-2081 45 13 eigenvector eigenvector NOUN ap-2081 45 14 of of ADP ap-2081 45 15 ĥ ĥ PUNCT ap-2081 45 16 with with ADP ap-2081 45 17 eigenvalue eigenvalue PROPN ap-2081 45 18 e e PROPN ap-2081 45 19 then then ADV ap-2081 45 20 â|ψ â|ψ PROPN ap-2081 45 21 〉 〉 PROPN ap-2081 45 22 is be AUX ap-2081 45 23 also also ADV ap-2081 45 24 an an DET ap-2081 45 25 eigenvector eigenvector NOUN ap-2081 45 26 with with ADP ap-2081 45 27 eigenvalue eigenvalue PROPN ap-2081 45 28 e∗. e∗. NOUN ap-2081 45 29 that that PRON ap-2081 45 30 is be AUX ap-2081 45 31 to to PART ap-2081 45 32 say say VERB ap-2081 45 33 : : PUNCT ap-2081 45 34 the the DET ap-2081 45 35 eigenvalues eigenvalue NOUN ap-2081 45 36 are be AUX ap-2081 45 37 either either CCONJ ap-2081 45 38 real real ADJ ap-2081 45 39 or or CCONJ ap-2081 45 40 appear appear VERB ap-2081 45 41 as as ADP ap-2081 45 42 pairs pair NOUN ap-2081 45 43 of of ADP ap-2081 45 44 complex complex ADJ ap-2081 45 45 conjugate conjugate ADJ ap-2081 45 46 numbers number NOUN ap-2081 45 47 . . PUNCT ap-2081 46 1 in in ADP ap-2081 46 2 the the DET ap-2081 46 3 former former ADJ ap-2081 46 4 case case NOUN ap-2081 46 5 ĥâ|ψ ĥâ|ψ VERB ap-2081 46 6 〉 〉 NOUN ap-2081 46 7 = = PUNCT ap-2081 46 8 eâ|ψ eâ|ψ PROPN ap-2081 46 9 〉 〉 NUM ap-2081 46 10 , , PUNCT ap-2081 46 11 ( ( PUNCT ap-2081 46 12 9 9 X ap-2081 46 13 ) ) PUNCT ap-2081 46 14 which which PRON ap-2081 46 15 contains contain VERB ap-2081 46 16 the the DET ap-2081 46 17 condition condition NOUN ap-2081 46 18 of of ADP ap-2081 46 19 unbroken unbroken ADJ ap-2081 46 20 symmetry symmetry NOUN ap-2081 46 21 [ [ X ap-2081 46 22 1 1 NUM ap-2081 46 23 ] ] PUNCT ap-2081 46 24 â|ψ â|ψ PROPN ap-2081 46 25 〉 〉 NOUN ap-2081 46 26 = = PUNCT ap-2081 46 27 λ|ψ λ|ψ ADV ap-2081 46 28 〉 〉 NOUN ap-2081 46 29 ( ( PUNCT ap-2081 46 30 10 10 NUM ap-2081 46 31 ) ) PUNCT ap-2081 46 32 as as ADP ap-2081 46 33 a a DET ap-2081 46 34 particular particular ADJ ap-2081 46 35 case case NOUN ap-2081 46 36 . . PUNCT ap-2081 47 1 note note VERB ap-2081 47 2 that that SCONJ ap-2081 47 3 equation equation NOUN ap-2081 47 4 ( ( PUNCT ap-2081 47 5 9 9 NUM ap-2081 47 6 ) ) PUNCT ap-2081 47 7 applies apply VERB ap-2081 47 8 to to ADP ap-2081 47 9 the the DET ap-2081 47 10 case case NOUN ap-2081 47 11 in in ADP ap-2081 47 12 which which PRON ap-2081 47 13 â|ψ â|ψ NOUN ap-2081 47 14 〉 〉 PROPN ap-2081 47 15 is be AUX ap-2081 47 16 a a DET ap-2081 47 17 linear linear ADJ ap-2081 47 18 combination combination NOUN ap-2081 47 19 of of ADP ap-2081 47 20 degenerate degenerate ADJ ap-2081 47 21 eigenvectors eigenvector NOUN ap-2081 47 22 of of ADP ap-2081 47 23 ĥ ĥ PUNCT ap-2081 47 24 with with ADP ap-2081 47 25 eigenvalue eigenvalue PROPN ap-2081 47 26 e. e. PROPN ap-2081 47 27 an an DET ap-2081 47 28 illustrative illustrative ADJ ap-2081 47 29 example example NOUN ap-2081 47 30 of of ADP ap-2081 47 31 this this DET ap-2081 47 32 more more ADV ap-2081 47 33 general general ADJ ap-2081 47 34 condition condition NOUN ap-2081 47 35 for for ADP ap-2081 47 36 real real ADJ ap-2081 47 37 eigenvalues eigenvalue NOUN ap-2081 47 38 is be AUX ap-2081 47 39 given give VERB ap-2081 47 40 elsewhere elsewhere ADV ap-2081 47 41 [ [ X ap-2081 47 42 5 5 NUM ap-2081 47 43 ] ] PUNCT ap-2081 47 44 . . PUNCT ap-2081 48 1 4 4 X ap-2081 48 2 . . X ap-2081 48 3 real real ADJ ap-2081 48 4 matrix matrix NOUN ap-2081 48 5 representation representation NOUN ap-2081 48 6 bender bender NOUN ap-2081 48 7 et et PROPN ap-2081 48 8 al al PROPN ap-2081 48 9 . . PUNCT ap-2081 49 1 [ [ X ap-2081 49 2 2 2 X ap-2081 49 3 ] ] PUNCT ap-2081 49 4 put put VERB ap-2081 49 5 forward forward ADV ap-2081 49 6 a a DET ap-2081 49 7 straightforward straightforward ADJ ap-2081 49 8 procedure procedure NOUN ap-2081 49 9 for for ADP ap-2081 49 10 obtaining obtain VERB ap-2081 49 11 a a DET ap-2081 49 12 basis basis NOUN ap-2081 49 13 set set VERB ap-2081 49 14 in in ADP ap-2081 49 15 which which PRON ap-2081 49 16 an an DET ap-2081 49 17 a a PRON ap-2081 49 18 - - PUNCT ap-2081 49 19 symmetric symmetric ADJ ap-2081 49 20 hamiltonian hamiltonian NOUN ap-2081 49 21 has have VERB ap-2081 49 22 a a DET ap-2081 49 23 real real ADJ ap-2081 49 24 matrix matrix NOUN ap-2081 49 25 representation representation NOUN ap-2081 49 26 . . PUNCT ap-2081 50 1 they they PRON ap-2081 50 2 proved prove VERB ap-2081 50 3 that that SCONJ ap-2081 50 4 for for ADP ap-2081 50 5 an an DET ap-2081 50 6 a a PRON ap-2081 50 7 - - PUNCT ap-2081 50 8 adapted adapt VERB ap-2081 50 9 basis basis NOUN ap-2081 50 10 set set NOUN ap-2081 50 11 { { PUNCT ap-2081 50 12 |na |na NUM ap-2081 50 13 〉 〉 NUM ap-2081 50 14 } } PUNCT ap-2081 50 15 â|na â|na NOUN ap-2081 50 16 〉 〉 NUM ap-2081 50 17 = = SYM ap-2081 50 18 |na |na PROPN ap-2081 50 19 〉 〉 NUM ap-2081 50 20 ( ( PUNCT ap-2081 50 21 11 11 NUM ap-2081 50 22 ) ) PUNCT ap-2081 50 23 the the DET ap-2081 50 24 matrix matrix NOUN ap-2081 50 25 elements element NOUN ap-2081 50 26 of of ADP ap-2081 50 27 the the DET ap-2081 50 28 invariant invariant ADJ ap-2081 50 29 hamiltonian hamiltonian ADJ ap-2081 50 30 operator operator NOUN ap-2081 50 31 are be AUX ap-2081 50 32 real real ADJ ap-2081 50 33 〈 〈 PROPN ap-2081 50 34 ma|ĥ|na ma|ĥ|na X ap-2081 50 35 〉 〉 NOUN ap-2081 50 36 = = SYM ap-2081 50 37 〈 〈 PROPN ap-2081 50 38 ma|ĥ|na〉∗ ma|ĥ|na〉∗ PROPN ap-2081 50 39 ( ( PUNCT ap-2081 50 40 12 12 NUM ap-2081 50 41 ) ) PUNCT ap-2081 50 42 these these DET ap-2081 50 43 authors author NOUN ap-2081 50 44 proposed propose VERB ap-2081 50 45 to to PART ap-2081 50 46 construct construct VERB ap-2081 50 47 |na |na NUM ap-2081 50 48 〉 〉 PROPN ap-2081 50 49 as as SCONJ ap-2081 50 50 ( ( PUNCT ap-2081 50 51 remember remember VERB ap-2081 50 52 that that SCONJ ap-2081 50 53 we we PRON ap-2081 50 54 have have AUX ap-2081 50 55 restricted restrict VERB ap-2081 50 56 present present ADJ ap-2081 50 57 discussion discussion NOUN ap-2081 50 58 to to ADP ap-2081 50 59 k k PROPN ap-2081 50 60 = = SYM ap-2081 50 61 1 1 NUM ap-2081 50 62 without without ADP ap-2081 50 63 loss loss NOUN ap-2081 50 64 of of ADP ap-2081 50 65 generality generality NOUN ap-2081 50 66 ) ) PUNCT ap-2081 50 67 |na |na NUM ap-2081 50 68 〉 〉 NOUN ap-2081 50 69 = = PUNCT ap-2081 50 70 |n〉+ |n〉+ ADJ ap-2081 50 71 â|n â|n PROPN ap-2081 50 72 〉 〉 NUM ap-2081 50 73 ( ( PUNCT ap-2081 50 74 13 13 NUM ap-2081 50 75 ) ) PUNCT ap-2081 50 76 where where SCONJ ap-2081 50 77 { { PUNCT ap-2081 50 78 |n |n NOUN ap-2081 50 79 〉 〉 NOUN ap-2081 50 80 } } PUNCT ap-2081 50 81 is be AUX ap-2081 50 82 any any DET ap-2081 50 83 orthonormal orthonormal ADJ ap-2081 50 84 basis basis NOUN ap-2081 50 85 set set NOUN ap-2081 50 86 . . PUNCT ap-2081 51 1 it it PRON ap-2081 51 2 is be AUX ap-2081 51 3 not not PART ap-2081 51 4 difficult difficult ADJ ap-2081 51 5 to to PART ap-2081 51 6 prove prove VERB ap-2081 51 7 that that SCONJ ap-2081 51 8 this this DET ap-2081 51 9 recipe recipe NOUN ap-2081 51 10 does do AUX ap-2081 51 11 not not PART ap-2081 51 12 apply apply VERB ap-2081 51 13 to to ADP ap-2081 51 14 any any DET ap-2081 51 15 basis basis NOUN ap-2081 51 16 set set NOUN ap-2081 51 17 . . PUNCT ap-2081 52 1 according accord VERB ap-2081 52 2 to to ADP ap-2081 52 3 equation equation NOUN ap-2081 52 4 ( ( PUNCT ap-2081 52 5 6 6 NUM ap-2081 52 6 ) ) PUNCT ap-2081 52 7 we we PRON ap-2081 52 8 can can AUX ap-2081 52 9 find find VERB ap-2081 52 10 a a DET ap-2081 52 11 basis basis NOUN ap-2081 52 12 set set NOUN ap-2081 52 13 { { PUNCT ap-2081 52 14 |n |n NOUN ap-2081 52 15 , , PUNCT ap-2081 52 16 σ σ PROPN ap-2081 52 17 〉 〉 PROPN ap-2081 52 18 } } PUNCT ap-2081 52 19 that that PRON ap-2081 52 20 satisfies satisfy VERB ap-2081 52 21 â|n â|n PROPN ap-2081 52 22 , , PUNCT ap-2081 52 23 σ σ PROPN ap-2081 52 24 〉 〉 NOUN ap-2081 52 25 = = SYM ap-2081 52 26 σ|n σ|n PROPN ap-2081 52 27 , , PUNCT ap-2081 52 28 σ σ PROPN ap-2081 52 29 〉 〉 PROPN ap-2081 52 30 , , PUNCT ap-2081 52 31 σ σ PROPN ap-2081 52 32 = = SYM ap-2081 52 33 ±1 ±1 VERB ap-2081 52 34 ( ( PUNCT ap-2081 52 35 14 14 NUM ap-2081 52 36 ) ) PUNCT ap-2081 52 37 consequently consequently ADV ap-2081 52 38 , , PUNCT ap-2081 52 39 all all DET ap-2081 52 40 the the DET ap-2081 52 41 vectors vector NOUN ap-2081 52 42 |n |n NOUN ap-2081 52 43 , , PUNCT ap-2081 52 44 σ〉a σ〉a PROPN ap-2081 52 45 = = SYM ap-2081 52 46 |n |n NOUN ap-2081 52 47 , , PUNCT ap-2081 52 48 σ〉+ σ〉+ PROPN ap-2081 52 49 â|n â|n PROPN ap-2081 52 50 , , PUNCT ap-2081 52 51 σ σ PROPN ap-2081 52 52 〉 〉 NUM ap-2081 52 53 = = SYM ap-2081 52 54 ( ( PUNCT ap-2081 52 55 1 1 NUM ap-2081 52 56 + + X ap-2081 52 57 σ)|n σ)|n NOUN ap-2081 52 58 , , PUNCT ap-2081 52 59 σ σ PROPN ap-2081 52 60 〉 〉 PROPN ap-2081 52 61 ( ( PUNCT ap-2081 52 62 15 15 NUM ap-2081 52 63 ) ) PUNCT ap-2081 52 64 with with ADP ap-2081 52 65 σ σ PROPN ap-2081 52 66 = = PUNCT ap-2081 52 67 −1 −1 NOUN ap-2081 52 68 vanish vanish VERB ap-2081 52 69 and and CCONJ ap-2081 52 70 the the DET ap-2081 52 71 resulting result VERB ap-2081 52 72 a a PRON ap-2081 52 73 - - PUNCT ap-2081 52 74 adapted adapt VERB ap-2081 52 75 vector vector NOUN ap-2081 52 76 set set NOUN ap-2081 52 77 is be AUX ap-2081 52 78 not not PART ap-2081 52 79 complete complete ADJ ap-2081 52 80 . . PUNCT ap-2081 53 1 we we PRON ap-2081 53 2 conclude conclude VERB ap-2081 53 3 that that SCONJ ap-2081 53 4 the the DET ap-2081 53 5 basis basis NOUN ap-2081 53 6 set set NOUN ap-2081 53 7 { { PUNCT ap-2081 53 8 |n |n NOUN ap-2081 53 9 〉 〉 NOUN ap-2081 53 10 } } PUNCT ap-2081 53 11 should should AUX ap-2081 53 12 be be AUX ap-2081 53 13 chosen choose VERB ap-2081 53 14 carefully carefully ADV ap-2081 53 15 in in ADP ap-2081 53 16 order order NOUN ap-2081 53 17 to to PART ap-2081 53 18 apply apply VERB ap-2081 53 19 the the DET ap-2081 53 20 recipe recipe NOUN ap-2081 53 21 of of ADP ap-2081 53 22 bender bender NOUN ap-2081 53 23 et et PROPN ap-2081 53 24 al al PROPN ap-2081 53 25 . . PUNCT ap-2081 54 1 [ [ X ap-2081 54 2 2 2 NUM ap-2081 54 3 ] ] PUNCT ap-2081 54 4 . . PUNCT ap-2081 55 1 in in ADP ap-2081 55 2 fact fact NOUN ap-2081 55 3 , , PUNCT ap-2081 55 4 the the DET ap-2081 55 5 authors author NOUN ap-2081 55 6 showed show VERB ap-2081 55 7 a a DET ap-2081 55 8 particular particular ADJ ap-2081 55 9 example example NOUN ap-2081 55 10 where where SCONJ ap-2081 55 11 it it PRON ap-2081 55 12 certainly certainly ADV ap-2081 55 13 applies apply VERB ap-2081 55 14 . . PUNCT ap-2081 56 1 we we PRON ap-2081 56 2 can can AUX ap-2081 56 3 construct construct VERB ap-2081 56 4 the the DET ap-2081 56 5 basis basis NOUN ap-2081 56 6 set set NOUN ap-2081 56 7 { { PUNCT ap-2081 56 8 |n |n NOUN ap-2081 56 9 , , PUNCT ap-2081 56 10 σ σ PROPN ap-2081 56 11 〉 〉 PROPN ap-2081 56 12 } } PUNCT ap-2081 56 13 from from ADP ap-2081 56 14 any any DET ap-2081 56 15 orthonormal orthonormal ADJ ap-2081 56 16 basis basis NOUN ap-2081 56 17 set set NOUN ap-2081 56 18 { { PUNCT ap-2081 56 19 |n |n NOUN ap-2081 56 20 〉 〉 NOUN ap-2081 56 21 } } PUNCT ap-2081 56 22 in in ADP ap-2081 56 23 the the DET ap-2081 56 24 following following ADJ ap-2081 56 25 way way NOUN ap-2081 56 26 |n |n NOUN ap-2081 56 27 , , PUNCT ap-2081 56 28 σ σ PROPN ap-2081 56 29 〉 〉 PROPN ap-2081 56 30 = = SYM ap-2081 56 31 nn nn PROPN ap-2081 56 32 , , PUNCT ap-2081 56 33 σq̂σ|n σq̂σ|n PROPN ap-2081 56 34 〉 〉 PROPN ap-2081 56 35 , , PUNCT ap-2081 56 36 q̂σ q̂σ NOUN ap-2081 56 37 = = SYM ap-2081 56 38 1 1 NUM ap-2081 56 39 2 2 NUM ap-2081 56 40 ( ( PUNCT ap-2081 56 41 1 1 NUM ap-2081 56 42 + + NUM ap-2081 56 43 σâ σâ NOUN ap-2081 56 44 ) ) PUNCT ap-2081 56 45 ( ( PUNCT ap-2081 56 46 16 16 NUM ap-2081 56 47 ) ) PUNCT ap-2081 56 48 where where SCONJ ap-2081 56 49 nn nn PROPN ap-2081 56 50 , , PUNCT ap-2081 56 51 σ σ PROPN ap-2081 56 52 is be AUX ap-2081 56 53 a a DET ap-2081 56 54 suitable suitable ADJ ap-2081 56 55 normalization normalization NOUN ap-2081 56 56 factor factor NOUN ap-2081 56 57 . . PUNCT ap-2081 57 1 it it PRON ap-2081 57 2 already already ADV ap-2081 57 3 satisfies satisfy VERB ap-2081 57 4 equation equation NOUN ap-2081 57 5 ( ( PUNCT ap-2081 57 6 14 14 NUM ap-2081 57 7 ) ) PUNCT ap-2081 57 8 because because SCONJ ap-2081 57 9 âq̂σ âq̂σ ADP ap-2081 57 10 = = PUNCT ap-2081 57 11 σq̂σ σq̂σ PROPN ap-2081 57 12 . . PUNCT ap-2081 58 1 in in ADP ap-2081 58 2 order order NOUN ap-2081 58 3 to to PART ap-2081 58 4 overcome overcome VERB ap-2081 58 5 the the DET ap-2081 58 6 shortcoming shortcoming NOUN ap-2081 58 7 in in ADP ap-2081 58 8 the the DET ap-2081 58 9 recipe recipe NOUN ap-2081 58 10 ( ( PUNCT ap-2081 58 11 13 13 NUM ap-2081 58 12 ) ) PUNCT ap-2081 58 13 we we PRON ap-2081 58 14 define define VERB ap-2081 58 15 the the DET ap-2081 58 16 a a PRON ap-2081 58 17 - - PUNCT ap-2081 58 18 adapted adapt VERB ap-2081 58 19 basis basis NOUN ap-2081 58 20 set set NOUN ap-2081 58 21 ba ba PROPN ap-2081 58 22 = = PRON ap-2081 58 23 { { PUNCT ap-2081 58 24 |n+ |n+ PROPN ap-2081 58 25 a a DET ap-2081 58 26 〉 〉 NOUN ap-2081 58 27 = = SYM ap-2081 58 28 |n |n NOUN ap-2081 58 29 , , PUNCT ap-2081 58 30 1 1 NUM ap-2081 58 31 〉 〉 NUM ap-2081 58 32 , , PUNCT ap-2081 58 33 |n−a |n−a ADV ap-2081 58 34 〉 〉 NOUN ap-2081 58 35 = = SYM ap-2081 58 36 i|n,−1 i|n,−1 ADJ ap-2081 58 37 〉 〉 NOUN ap-2081 58 38 } } PUNCT ap-2081 58 39 . . PUNCT ap-2081 59 1 note note VERB ap-2081 59 2 that that SCONJ ap-2081 59 3 the the DET ap-2081 59 4 vectors vector NOUN ap-2081 59 5 |n±a |n±a VERB ap-2081 59 6 〉 〉 PROPN ap-2081 59 7 satisfy satisfy VERB ap-2081 59 8 the the DET ap-2081 59 9 requirement requirement NOUN ap-2081 59 10 ( ( PUNCT ap-2081 59 11 11 11 NUM ap-2081 59 12 ) ) PUNCT ap-2081 59 13 and and CCONJ ap-2081 59 14 that that SCONJ ap-2081 59 15 ba ba PROPN ap-2081 59 16 is be AUX ap-2081 59 17 complete complete ADJ ap-2081 59 18 . . PUNCT ap-2081 60 1 in in ADP ap-2081 60 2 principle principle NOUN ap-2081 60 3 there there PRON ap-2081 60 4 is be VERB ap-2081 60 5 no no DET ap-2081 60 6 guarantee guarantee NOUN ap-2081 60 7 of of ADP ap-2081 60 8 orthogonality orthogonality NOUN ap-2081 60 9 , , PUNCT ap-2081 60 10 but but CCONJ ap-2081 60 11 such such DET ap-2081 60 12 a a DET ap-2081 60 13 difficulty difficulty NOUN ap-2081 60 14 does do AUX ap-2081 60 15 not not PART ap-2081 60 16 arise arise VERB ap-2081 60 17 in in ADP ap-2081 60 18 the the DET ap-2081 60 19 examples example NOUN ap-2081 60 20 discussed discuss VERB ap-2081 60 21 below below ADV ap-2081 60 22 . . PUNCT ap-2081 61 1 the the DET ap-2081 61 2 vectors vector NOUN ap-2081 61 3 |v±n |v±n NOUN ap-2081 61 4 〉 〉 NOUN ap-2081 61 5 = = SYM ap-2081 61 6 1√ 1√ PROPN ap-2081 61 7 2 2 NUM ap-2081 61 8 ( ( PUNCT ap-2081 61 9 |n+ |n+ PROPN ap-2081 61 10 a a DET ap-2081 61 11 〉 〉 PROPN ap-2081 61 12 ± ± NUM ap-2081 61 13 |n |n NOUN ap-2081 61 14 − − PROPN ap-2081 61 15 a a DET ap-2081 61 16 〉 〉 NOUN ap-2081 61 17 ) ) PUNCT ap-2081 61 18 also also ADV ap-2081 61 19 satisfy satisfy VERB ap-2081 61 20 the the DET ap-2081 61 21 requirement requirement NOUN ap-2081 61 22 ( ( PUNCT ap-2081 61 23 11 11 NUM ap-2081 61 24 ) ) PUNCT ap-2081 61 25 and and CCONJ ap-2081 61 26 in in ADP ap-2081 61 27 the the DET ap-2081 61 28 particular particular ADJ ap-2081 61 29 case case NOUN ap-2081 61 30 of of ADP ap-2081 61 31 a a DET ap-2081 61 32 two two NUM ap-2081 61 33 - - PUNCT ap-2081 61 34 dimensional dimensional ADJ ap-2081 61 35 space space NOUN ap-2081 61 36 they they PRON ap-2081 61 37 lead lead VERB ap-2081 61 38 to to ADP ap-2081 61 39 the the DET ap-2081 61 40 a a PRON ap-2081 61 41 - - PUNCT ap-2081 61 42 adapted adapt VERB ap-2081 61 43 basis basis NOUN ap-2081 61 44 set set NOUN ap-2081 61 45 chosen choose VERB ap-2081 61 46 by by ADP ap-2081 61 47 bender bender PROPN ap-2081 61 48 et et PROPN ap-2081 61 49 al al PROPN ap-2081 61 50 . . PUNCT ap-2081 62 1 [ [ X ap-2081 62 2 2 2 X ap-2081 62 3 ] ] PUNCT ap-2081 62 4 to to PART ap-2081 62 5 introduce introduce VERB ap-2081 62 6 the the DET ap-2081 62 7 issue issue NOUN ap-2081 62 8 by by ADP ap-2081 62 9 means mean NOUN ap-2081 62 10 of of ADP ap-2081 62 11 a a DET ap-2081 62 12 simple simple ADJ ap-2081 62 13 example example NOUN ap-2081 62 14 . . PUNCT ap-2081 63 1 as as SCONJ ap-2081 63 2 a a DET ap-2081 63 3 particular particular ADJ ap-2081 63 4 case case NOUN ap-2081 63 5 consider consider VERB ap-2081 63 6 the the DET ap-2081 63 7 parity parity NOUN ap-2081 63 8 - - PUNCT ap-2081 63 9 time time NOUN ap-2081 63 10 antiunitary antiunitary NOUN ap-2081 63 11 operator operator NOUN ap-2081 63 12 â â X ap-2081 64 1 = = SYM ap-2081 64 2 p̂ p̂ NOUN ap-2081 64 3 t̂ t̂ NUM ap-2081 64 4 , , PUNCT ap-2081 64 5 where where SCONJ ap-2081 64 6 p̂ p̂ NOUN ap-2081 64 7 an an DET ap-2081 64 8 t̂ t̂ NUM ap-2081 64 9 are be AUX ap-2081 64 10 the the DET ap-2081 64 11 parity parity NOUN ap-2081 64 12 and and CCONJ ap-2081 64 13 time time NOUN ap-2081 64 14 - - PUNCT ap-2081 64 15 reversal reversal NOUN ap-2081 64 16 operators operator NOUN ap-2081 64 17 , , PUNCT ap-2081 64 18 respectively respectively ADV ap-2081 64 19 [ [ X ap-2081 64 20 3 3 NUM ap-2081 64 21 ] ] PUNCT ap-2081 64 22 . . PUNCT ap-2081 65 1 let let VERB ap-2081 65 2 { { PUNCT ap-2081 65 3 |n |n NOUN ap-2081 65 4 〉 〉 NOUN ap-2081 65 5 , , PUNCT ap-2081 65 6 n n NOUN ap-2081 65 7 = = SYM ap-2081 65 8 0 0 NUM ap-2081 65 9 , , PUNCT ap-2081 65 10 1 1 NUM ap-2081 65 11 , , PUNCT ap-2081 65 12 . . PUNCT ap-2081 65 13 . . PUNCT ap-2081 66 1 . . PUNCT ap-2081 66 2 } } PUNCT ap-2081 66 3 be be AUX ap-2081 66 4 the the DET ap-2081 66 5 basis basis NOUN ap-2081 66 6 set set NOUN ap-2081 66 7 of of ADP ap-2081 66 8 eigenvectors eigenvector NOUN ap-2081 66 9 of of ADP ap-2081 66 10 the the DET ap-2081 66 11 harmonic harmonic ADJ ap-2081 66 12 oscillator oscillator NOUN ap-2081 66 13 ĥ0 ĥ0 PUNCT ap-2081 66 14 = = PUNCT ap-2081 67 1 p̂2 p̂2 NOUN ap-2081 68 1 + + CCONJ ap-2081 68 2 x̂2 x̂2 NOUN ap-2081 68 3 that that PRON ap-2081 68 4 are be AUX ap-2081 68 5 real real ADJ ap-2081 68 6 and and CCONJ ap-2081 68 7 satisfy satisfy VERB ap-2081 68 8 p̂ p̂ NOUN ap-2081 68 9 |n |n NOUN ap-2081 68 10 〉 〉 NOUN ap-2081 68 11 = = SYM ap-2081 68 12 ( ( PUNCT ap-2081 68 13 −1)n|n −1)n|n NOUN ap-2081 68 14 〉 〉 NOUN ap-2081 69 1 so so SCONJ ap-2081 69 2 that that SCONJ ap-2081 69 3 â|n â|n NOUN ap-2081 69 4 〉 〉 NOUN ap-2081 69 5 = = SYM ap-2081 69 6 ( ( PUNCT ap-2081 69 7 −1)n|n −1)n|n NOUN ap-2081 69 8 〉 〉 NUM ap-2081 69 9 . . PUNCT ap-2081 70 1 it it PRON ap-2081 70 2 is be AUX ap-2081 70 3 clear clear ADJ ap-2081 70 4 that that SCONJ ap-2081 70 5 the the DET ap-2081 70 6 recipe recipe NOUN ap-2081 70 7 ( ( PUNCT ap-2081 70 8 13 13 NUM ap-2081 70 9 ) ) PUNCT ap-2081 70 10 does do AUX ap-2081 70 11 not not PART ap-2081 70 12 apply apply VERB ap-2081 70 13 to to ADP ap-2081 70 14 this this DET ap-2081 70 15 simple simple ADJ ap-2081 70 16 case case NOUN ap-2081 70 17 . . PUNCT ap-2081 71 1 on on ADP ap-2081 71 2 the the DET ap-2081 71 3 other other ADJ ap-2081 71 4 hand hand NOUN ap-2081 71 5 , , PUNCT ap-2081 71 6 the the DET ap-2081 71 7 present present ADJ ap-2081 71 8 recipe recipe NOUN ap-2081 71 9 yields yield VERB ap-2081 71 10 the the DET ap-2081 71 11 a a PRON ap-2081 71 12 - - PUNCT ap-2081 71 13 adapted adapt VERB ap-2081 71 14 basis basis NOUN ap-2081 71 15 set set NOUN ap-2081 71 16 bhoa bhoa NOUN ap-2081 71 17 = = SYM ap-2081 71 18 { { PUNCT ap-2081 71 19 |2n |2n PROPN ap-2081 71 20 〉 〉 NUM ap-2081 71 21 , , PUNCT ap-2081 71 22 i|2n i|2n PROPN ap-2081 71 23 + + PROPN ap-2081 71 24 1 1 NUM ap-2081 71 25 〉 〉 NUM ap-2081 71 26 , , PUNCT ap-2081 71 27 n n NOUN ap-2081 71 28 = = SYM ap-2081 71 29 0 0 NUM ap-2081 71 30 , , PUNCT ap-2081 71 31 1 1 NUM ap-2081 71 32 , , PUNCT ap-2081 71 33 . . PUNCT ap-2081 71 34 . . PUNCT ap-2081 72 1 . . PUNCT ap-2081 72 2 } } PUNCT ap-2081 73 1 which which PRON ap-2081 73 2 is be AUX ap-2081 73 3 obviously obviously ADV ap-2081 73 4 complete complete ADJ ap-2081 73 5 . . PUNCT ap-2081 74 1 every every DET ap-2081 74 2 vector vector NOUN ap-2081 74 3 of of ADP ap-2081 74 4 the the DET ap-2081 74 5 orthonormal orthonormal ADJ ap-2081 74 6 basis basis NOUN ap-2081 74 7 set set NOUN ap-2081 74 8 bhoa bhoa NOUN ap-2081 74 9 in in ADP ap-2081 74 10 the the DET ap-2081 74 11 coordinate coordinate NOUN ap-2081 74 12 representation representation NOUN ap-2081 74 13 can can AUX ap-2081 74 14 be be AUX ap-2081 74 15 expressed express VERB ap-2081 74 16 as as ADP ap-2081 74 17 a a DET ap-2081 74 18 linear linear ADJ ap-2081 74 19 combination combination NOUN ap-2081 74 20 of of ADP ap-2081 74 21 the the DET ap-2081 74 22 elements element NOUN ap-2081 74 23 of of ADP ap-2081 74 24 the the DET ap-2081 74 25 nonorthogonal nonorthogonal ADJ ap-2081 74 26 basis basis NOUN ap-2081 74 27 set set NOUN ap-2081 74 28 { { PUNCT ap-2081 74 29 fn(x fn(x PRON ap-2081 74 30 ) ) PUNCT ap-2081 74 31 = = SYM ap-2081 75 1 e−x e−x PROPN ap-2081 75 2 2/2(ix)n 2/2(ix)n NUM ap-2081 75 3 , , PUNCT ap-2081 75 4 n n NOUN ap-2081 75 5 = = SYM ap-2081 75 6 0 0 NUM ap-2081 75 7 , , PUNCT ap-2081 75 8 1 1 NUM ap-2081 75 9 , , PUNCT ap-2081 75 10 . . PUNCT ap-2081 75 11 . . PUNCT ap-2081 75 12 . . PUNCT ap-2081 75 13 } } PUNCT ap-2081 75 14 . . PUNCT ap-2081 76 1 by by ADP ap-2081 76 2 means mean NOUN ap-2081 76 3 of of ADP ap-2081 76 4 a a DET ap-2081 76 5 slight slight ADJ ap-2081 76 6 generalization generalization NOUN ap-2081 76 7 of of ADP ap-2081 76 8 the the DET ap-2081 76 9 latter latter ADJ ap-2081 76 10 , , PUNCT ap-2081 76 11 znojil znojil NOUN ap-2081 77 1 [ [ X ap-2081 77 2 6 6 NUM ap-2081 77 3 ] ] PUNCT ap-2081 77 4 derived derive VERB ap-2081 77 5 a a DET ap-2081 77 6 recurrence recurrence NOUN ap-2081 77 7 relation relation NOUN ap-2081 77 8 with with ADP ap-2081 77 9 real real ADJ ap-2081 77 10 coefficients coefficient NOUN ap-2081 77 11 for for ADP ap-2081 77 12 a a DET ap-2081 77 13 family family NOUN ap-2081 77 14 of of ADP ap-2081 77 15 complex complex ADJ ap-2081 77 16 anharmonic anharmonic ADJ ap-2081 77 17 potentials potential NOUN ap-2081 77 18 . . PUNCT ap-2081 78 1 he he PRON ap-2081 78 2 also also ADV ap-2081 78 3 constructed construct VERB ap-2081 78 4 a a DET ap-2081 78 5 real real ADJ ap-2081 78 6 matrix matrix NOUN ap-2081 78 7 representation representation NOUN ap-2081 78 8 of of ADP ap-2081 78 9 a a DET ap-2081 78 10 ptsymmetric ptsymmetric ADJ ap-2081 78 11 oscillator oscillator NOUN ap-2081 78 12 in in ADP ap-2081 78 13 terms term NOUN ap-2081 78 14 of of ADP ap-2081 78 15 the the DET ap-2081 78 16 eigenvectors eigenvector NOUN ap-2081 78 17 of of ADP ap-2081 78 18 â â X ap-2081 78 19 = = NOUN ap-2081 78 20 p̂ p̂ NOUN ap-2081 78 21 t̂ t̂ PUNCT ap-2081 79 1 [ [ X ap-2081 79 2 7 7 NUM ap-2081 79 3 ] ] PUNCT ap-2081 79 4 . . PUNCT ap-2081 80 1 note note VERB ap-2081 80 2 that that SCONJ ap-2081 80 3 his his PRON ap-2081 80 4 vectors vector NOUN ap-2081 80 5 |sn |sn VERB ap-2081 80 6 〉 〉 NUM ap-2081 80 7 and and CCONJ ap-2081 80 8 |ln |ln PROPN ap-2081 80 9 〉 〉 NOUN ap-2081 80 10 are be AUX ap-2081 80 11 our our PRON ap-2081 80 12 |n |n NOUN ap-2081 80 13 , , PUNCT ap-2081 80 14 1 1 NUM ap-2081 80 15 〉 〉 NUM ap-2081 80 16 and and CCONJ ap-2081 80 17 |n,−1 |n,−1 PROPN ap-2081 80 18 〉 〉 PROPN ap-2081 80 19 respectively respectively ADV ap-2081 80 20 . . PUNCT ap-2081 81 1 following follow VERB ap-2081 81 2 porter porter NOUN ap-2081 81 3 [ [ X ap-2081 81 4 3 3 X ap-2081 81 5 ] ] PUNCT ap-2081 81 6 we we PRON ap-2081 81 7 can can AUX ap-2081 81 8 try try VERB ap-2081 81 9 the the DET ap-2081 81 10 ansatz ansatz ADJ ap-2081 81 11 |na |na NOUN ap-2081 81 12 〉 〉 NOUN ap-2081 81 13 = = PUNCT ap-2081 82 1 an|n〉+ an|n〉+ ADP ap-2081 82 2 âan|n âan|n NOUN ap-2081 82 3 〉 〉 NOUN ap-2081 82 4 = = PUNCT ap-2081 83 1 an|n〉+ an|n〉+ ADP ap-2081 83 2 a∗nâ|n a∗nâ|n PROPN ap-2081 83 3 〉 〉 PROPN ap-2081 83 4 ( ( PUNCT ap-2081 83 5 17 17 NUM ap-2081 83 6 ) ) PUNCT ap-2081 83 7 which which PRON ap-2081 83 8 already already ADV ap-2081 83 9 satisfies satisfy VERB ap-2081 83 10 â|na â|na NOUN ap-2081 83 11 〉 〉 PROPN ap-2081 83 12 = = SYM ap-2081 83 13 |na |na PROPN ap-2081 83 14 〉 〉 NUM ap-2081 83 15 . . PUNCT ap-2081 84 1 this this DET ap-2081 84 2 definition definition NOUN ap-2081 84 3 of of ADP ap-2081 84 4 an an DET ap-2081 84 5 a a PRON ap-2081 84 6 - - PUNCT ap-2081 84 7 adapted adapt VERB ap-2081 84 8 basis basis NOUN ap-2081 84 9 set set NOUN ap-2081 84 10 is be AUX ap-2081 84 11 slightly slightly ADV ap-2081 84 12 more more ADV ap-2081 84 13 general general ADJ ap-2081 84 14 than than ADP ap-2081 84 15 equation equation NOUN ap-2081 84 16 ( ( PUNCT ap-2081 84 17 13 13 NUM ap-2081 84 18 ) ) PUNCT ap-2081 84 19 . . PUNCT ap-2081 85 1 when when SCONJ ap-2081 85 2 â|n â|n PROPN ap-2081 85 3 〉 〉 NUM ap-2081 85 4 = = SYM ap-2081 85 5 ( ( PUNCT ap-2081 85 6 −1)n|n −1)n|n NOUN ap-2081 85 7 〉 〉 NOUN ap-2081 85 8 we we PRON ap-2081 85 9 simply simply ADV ap-2081 85 10 choose choose VERB ap-2081 85 11 an an DET ap-2081 85 12 = = SYM ap-2081 85 13 1 1 NUM ap-2081 85 14 2 2 NUM ap-2081 85 15 ( ( PUNCT ap-2081 85 16 1 1 NUM ap-2081 85 17 + + NUM ap-2081 85 18 i i NOUN ap-2081 85 19 ) ) PUNCT ap-2081 85 20 and and CCONJ ap-2081 85 21 obtain obtain VERB ap-2081 85 22 the the DET ap-2081 85 23 result result NOUN ap-2081 85 24 above above ADP ap-2081 85 25 for for ADP ap-2081 85 26 the the DET ap-2081 85 27 particular particular ADJ ap-2081 85 28 case case NOUN ap-2081 85 29 of of ADP ap-2081 85 30 the the DET ap-2081 85 31 harmonic harmonic ADJ ap-2081 85 32 - - PUNCT ap-2081 85 33 oscillator oscillator NOUN ap-2081 85 34 basis basis NOUN ap-2081 85 35 set set NOUN ap-2081 85 36 . . PUNCT ap-2081 86 1 note note VERB ap-2081 86 2 that that SCONJ ap-2081 86 3 the the DET ap-2081 86 4 resulting result VERB ap-2081 86 5 expressions expression NOUN ap-2081 86 6 ( ( PUNCT ap-2081 86 7 we we PRON ap-2081 86 8 can can AUX ap-2081 86 9 also also ADV ap-2081 86 10 choose choose VERB ap-2081 86 11 an an DET ap-2081 86 12 = = SYM ap-2081 86 13 1 1 NUM ap-2081 86 14 2 2 NUM ap-2081 86 15 ( ( PUNCT ap-2081 86 16 1− 1− NUM ap-2081 86 17 i i NOUN ap-2081 86 18 ) ) PUNCT ap-2081 86 19 ) ) PUNCT ap-2081 86 20 are be AUX ap-2081 86 21 similar similar ADJ ap-2081 86 22 to to ADP ap-2081 86 23 those those PRON ap-2081 86 24 in in ADP ap-2081 86 25 equation equation NOUN ap-2081 86 26 ( ( PUNCT ap-2081 86 27 16 16 NUM ap-2081 86 28 ) ) PUNCT ap-2081 86 29 in in ADP ap-2081 86 30 the the DET ap-2081 86 31 paper paper NOUN ap-2081 86 32 of of ADP ap-2081 86 33 bender bender PROPN ap-2081 86 34 et et PROPN ap-2081 86 35 al al PROPN ap-2081 86 36 . . PUNCT ap-2081 87 1 [ [ X ap-2081 87 2 2 2 NUM ap-2081 87 3 ] ] PUNCT ap-2081 87 4 . . PUNCT ap-2081 88 1 114 114 NUM ap-2081 88 2 vol vol NOUN ap-2081 88 3 . . PUNCT ap-2081 89 1 54 54 NUM ap-2081 89 2 no no NOUN ap-2081 89 3 . . PUNCT ap-2081 90 1 2/2014 2/2014 NUM ap-2081 90 2 real real ADJ ap-2081 90 3 matrix matrix NOUN ap-2081 90 4 representation representation NOUN ap-2081 90 5 5 5 NUM ap-2081 90 6 . . PUNCT ap-2081 91 1 the the DET ap-2081 91 2 harmonic harmonic ADJ ap-2081 91 3 - - PUNCT ap-2081 91 4 oscillator oscillator NOUN ap-2081 91 5 basis basis NOUN ap-2081 91 6 set set VERB ap-2081 91 7 many many ADJ ap-2081 91 8 examples example NOUN ap-2081 91 9 of of ADP ap-2081 91 10 pt pt NOUN ap-2081 91 11 -symmetric -symmetric ADJ ap-2081 91 12 hamiltonians hamiltonian NOUN ap-2081 91 13 are be AUX ap-2081 91 14 one one NUM ap-2081 91 15 - - PUNCT ap-2081 91 16 dimensional dimensional ADJ ap-2081 91 17 models model NOUN ap-2081 91 18 of of ADP ap-2081 91 19 the the DET ap-2081 91 20 form form NOUN ap-2081 91 21 [ [ X ap-2081 91 22 1 1 NUM ap-2081 91 23 ] ] PUNCT ap-2081 91 24 ĥ ĥ X ap-2081 91 25 = = PUNCT ap-2081 92 1 p̂2 p̂2 PROPN ap-2081 92 2 + + CCONJ ap-2081 92 3 v v X ap-2081 92 4 ( ( PUNCT ap-2081 92 5 x x NOUN ap-2081 92 6 ) ) PUNCT ap-2081 92 7 , , PUNCT ap-2081 92 8 ( ( PUNCT ap-2081 92 9 18 18 NUM ap-2081 92 10 ) ) PUNCT ap-2081 92 11 where where SCONJ ap-2081 92 12 v v X ap-2081 92 13 ( ( PUNCT ap-2081 92 14 −x)∗ −x)∗ NOUN ap-2081 92 15 = = SYM ap-2081 92 16 v v NOUN ap-2081 92 17 ( ( PUNCT ap-2081 92 18 x x NOUN ap-2081 92 19 ) ) PUNCT ap-2081 92 20 . . PUNCT ap-2081 93 1 ( ( PUNCT ap-2081 93 2 19 19 NUM ap-2081 93 3 ) ) PUNCT ap-2081 93 4 we we PRON ap-2081 93 5 can can AUX ap-2081 93 6 write write VERB ap-2081 93 7 v v NOUN ap-2081 93 8 ( ( PUNCT ap-2081 93 9 x x NOUN ap-2081 93 10 ) ) PUNCT ap-2081 93 11 as as ADP ap-2081 93 12 the the DET ap-2081 93 13 sum sum NOUN ap-2081 93 14 of of ADP ap-2081 93 15 its its PRON ap-2081 93 16 even even ADJ ap-2081 93 17 ve(−x ve(−x NOUN ap-2081 93 18 ) ) PUNCT ap-2081 93 19 = = SYM ap-2081 93 20 ve(x ve(x NOUN ap-2081 93 21 ) ) PUNCT ap-2081 93 22 and and CCONJ ap-2081 93 23 odd odd ADJ ap-2081 93 24 vo(−x vo(−x NOUN ap-2081 93 25 ) ) PUNCT ap-2081 93 26 = = SYM ap-2081 93 27 −vo(x −vo(x NOUN ap-2081 93 28 ) ) PUNCT ap-2081 93 29 parts part NOUN ap-2081 93 30 v v NOUN ap-2081 93 31 ( ( PUNCT ap-2081 93 32 x x NOUN ap-2081 93 33 ) ) PUNCT ap-2081 93 34 = = SYM ap-2081 93 35 ve(x ve(x NOUN ap-2081 93 36 ) ) PUNCT ap-2081 93 37 + + CCONJ ap-2081 93 38 vo(x vo(x NUM ap-2081 93 39 ) ) PUNCT ap-2081 93 40 , , PUNCT ap-2081 93 41 ( ( PUNCT ap-2081 93 42 20 20 NUM ap-2081 93 43 ) ) PUNCT ap-2081 94 1 where where SCONJ ap-2081 94 2 ve(x ve(x NOUN ap-2081 94 3 ) ) PUNCT ap-2081 94 4 = = SYM ap-2081 94 5 1 1 NUM ap-2081 94 6 2 2 NUM ap-2081 94 7 [ [ PUNCT ap-2081 94 8 v v NOUN ap-2081 94 9 ( ( PUNCT ap-2081 94 10 x x NOUN ap-2081 94 11 ) ) PUNCT ap-2081 94 12 + + CCONJ ap-2081 94 13 v v X ap-2081 94 14 ( ( PUNCT ap-2081 94 15 −x −x NOUN ap-2081 94 16 ) ) PUNCT ap-2081 94 17 ] ] PUNCT ap-2081 95 1 = = PUNCT ap-2081 95 2 < < X ap-2081 95 3 v v X ap-2081 95 4 ( ( PUNCT ap-2081 95 5 x x NOUN ap-2081 95 6 ) ) PUNCT ap-2081 95 7 , , PUNCT ap-2081 95 8 vo(x vo(x X ap-2081 95 9 ) ) PUNCT ap-2081 95 10 = = SYM ap-2081 95 11 1 1 NUM ap-2081 95 12 2 2 NUM ap-2081 95 13 [ [ PUNCT ap-2081 95 14 v v NOUN ap-2081 95 15 ( ( PUNCT ap-2081 95 16 x)− x)− PROPN ap-2081 95 17 v v PROPN ap-2081 95 18 ( ( PUNCT ap-2081 95 19 −x −x NOUN ap-2081 95 20 ) ) PUNCT ap-2081 95 21 ] ] PUNCT ap-2081 96 1 = = PUNCT ap-2081 97 1 i i PRON ap-2081 97 2 = = NOUN ap-2081 97 3 v v X ap-2081 97 4 ( ( PUNCT ap-2081 97 5 x x NOUN ap-2081 97 6 ) ) PUNCT ap-2081 97 7 . . PUNCT ap-2081 98 1 ( ( PUNCT ap-2081 98 2 21 21 NUM ap-2081 98 3 ) ) PUNCT ap-2081 98 4 for for ADP ap-2081 98 5 convenience convenience NOUN ap-2081 98 6 we we PRON ap-2081 98 7 change change VERB ap-2081 98 8 the the DET ap-2081 98 9 notation notation NOUN ap-2081 98 10 of of ADP ap-2081 98 11 the the DET ap-2081 98 12 preceding precede VERB ap-2081 98 13 section section NOUN ap-2081 98 14 and and CCONJ ap-2081 98 15 define define VERB ap-2081 98 16 the the DET ap-2081 98 17 a a PRON ap-2081 98 18 - - PUNCT ap-2081 98 19 adapted adapt VERB ap-2081 98 20 basis basis NOUN ap-2081 98 21 set set NOUN ap-2081 98 22 { { PUNCT ap-2081 98 23 ϕn ϕn NOUN ap-2081 98 24 } } PUNCT ap-2081 98 25 as as ADP ap-2081 98 26 |ϕ2n |ϕ2n NOUN ap-2081 98 27 〉 〉 NOUN ap-2081 98 28 = = SYM ap-2081 98 29 |2n |2n PROPN ap-2081 98 30 〉 〉 PROPN ap-2081 98 31 |ϕ2n+1 |ϕ2n+1 PROPN ap-2081 98 32 〉 〉 NOUN ap-2081 98 33 = = SYM ap-2081 98 34 i|2n+ i|2n+ VERB ap-2081 98 35 1 1 NUM ap-2081 98 36 〉 〉 NUM ap-2081 98 37 , , PUNCT ap-2081 98 38 n n NOUN ap-2081 98 39 = = SYM ap-2081 98 40 0 0 NUM ap-2081 98 41 , , PUNCT ap-2081 98 42 1 1 NUM ap-2081 98 43 , , PUNCT ap-2081 98 44 . . PUNCT ap-2081 98 45 . . PUNCT ap-2081 98 46 . . PUNCT ap-2081 99 1 , , PUNCT ap-2081 99 2 ( ( PUNCT ap-2081 99 3 22 22 NUM ap-2081 99 4 ) ) PUNCT ap-2081 99 5 where where SCONJ ap-2081 99 6 { { PUNCT ap-2081 99 7 |n |n NOUN ap-2081 99 8 〉 〉 NOUN ap-2081 99 9 } } PUNCT ap-2081 99 10 is be AUX ap-2081 99 11 the the DET ap-2081 99 12 harmonic harmonic ADJ ap-2081 99 13 - - PUNCT ap-2081 99 14 oscillator oscillator NOUN ap-2081 99 15 basis basis NOUN ap-2081 99 16 set set NOUN ap-2081 99 17 . . PUNCT ap-2081 100 1 therefore therefore ADV ap-2081 100 2 〈 〈 PROPN ap-2081 100 3 ϕ2n|p̂2|ϕ2 ϕ2n|p̂2|ϕ2 NOUN ap-2081 100 4 m m NOUN ap-2081 100 5 〉 〉 NOUN ap-2081 100 6 = = SYM ap-2081 100 7 〈 〈 PROPN ap-2081 100 8 φ2n|p̂2|φ2 φ2n|p̂2|φ2 NOUN ap-2081 100 9 m m NOUN ap-2081 100 10 〉 〉 NOUN ap-2081 100 11 〈 〈 PROPN ap-2081 100 12 ϕ2n|p̂2|ϕ2m+1 ϕ2n|p̂2|ϕ2m+1 PROPN ap-2081 100 13 〉 〉 PROPN ap-2081 100 14 = = SYM ap-2081 100 15 〈 〈 PROPN ap-2081 100 16 φ2n|p̂2|φ2m+1 φ2n|p̂2|φ2m+1 ADJ ap-2081 100 17 〉 〉 PROPN ap-2081 100 18 = = SYM ap-2081 100 19 0 0 NUM ap-2081 100 20 〈 〈 PROPN ap-2081 100 21 ϕ2m+1|p̂2|ϕ2n ϕ2m+1|p̂2|ϕ2n PROPN ap-2081 100 22 〉 〉 NOUN ap-2081 100 23 = = SYM ap-2081 100 24 〈 〈 PROPN ap-2081 100 25 φ2m+1|p̂2|φ2n φ2m+1|p̂2|φ2n PROPN ap-2081 100 26 〉 〉 PROPN ap-2081 100 27 = = SYM ap-2081 100 28 0 0 NUM ap-2081 101 1 〈 〈 PROPN ap-2081 101 2 ϕ2n+1|p̂2|ϕ2m+1 ϕ2n+1|p̂2|ϕ2m+1 PROPN ap-2081 101 3 〉 〉 PROPN ap-2081 101 4 = = SYM ap-2081 101 5 〈 〈 PROPN ap-2081 101 6 φ2n+1|p̂2|φ2m+1 φ2n+1|p̂2|φ2m+1 ADJ ap-2081 101 7 〉 〉 PROPN ap-2081 101 8 ( ( PUNCT ap-2081 101 9 23 23 NUM ap-2081 101 10 ) ) PUNCT ap-2081 101 11 and and CCONJ ap-2081 101 12 〈 〈 PROPN ap-2081 101 13 ϕ2n|v ϕ2n|v PROPN ap-2081 101 14 |ϕ2 |ϕ2 NOUN ap-2081 101 15 m m NOUN ap-2081 101 16 〉 〉 NOUN ap-2081 101 17 = = SYM ap-2081 101 18 〈 〈 PROPN ap-2081 101 19 φ2n|