New Version of the Rayleigh–Schrdinger Perturbation Theory(9)

ABSTRACT: It has been shown in our preceding papers that the linear dependence of the perturbation wave functions on the perturbation energies makes possible to calculate the exact perturbation energies from the values of the perturbation wave functions co

TABLEIV_____________________________________________________________________________________________

PerturbationenergiesEnforthegroundstateoftheHe´non–HeilesHamiltonian(31,32);E0istheexactzero-*orderenergy.n0

1.65

2

0.338300661

4

0.95061029

6

5.26789699

81012141618202224262830

41.495774 414.584851 4969.6014 69209.077 1096653.64 0.194809520 108 0.38361825 109 0.82989680 1010 0.19576261 1012 0.50030730 1013 0.13776229 1015 0.40671279 1016

numEn

anEn

33

1.6520

4096775

0.338300661345...

1056557053570119375

0.950610298642...

1111451301420501184

1316035702524299260768279480015625

5.26789699698...

58464282879382983469996471322587023716733119140625

41.4957745445...

1408921354549645735897393295254864019710396022784 414.584851968... 4969.60146844... 69209.0774461... 1096653.64936...

numan*TheenergiescalculatedfromEq.(12)aredenotedasEn.TheanalyticenergiesdenotedasEnwerecalculatedforn 0,2,...,

anananan

16.Forlackofspace,theanalyticenergiesE10,E12,E14,andE16arewrittenasadecimalnumberonly.Odd-orderperturbationenergiesequalzero.

TABLEV______________________________________________________________________________________________

PerturbationenergiesEnforthe rstexcited[0,1]stateoftheHe´non–HeilesHamiltonian(31,32);E0istheexactzero-orderenergy.*n0

2.95

2

2.649761

4

13.663991

681012141618202224262830

130.211814 1656.04703 25297.5508 441706.519 0.85765206 107 0.1821097391 109 0.418283241 1010 0.103158272 1012 0.271740662 1013 0.761625748 1014 0.22644751 1016 0.712486581 1017 0.23673444 1019

numEn

anEn

59

2.95866382925

2.64976191982...

326966328

298922976349768528158125

13.6639909800...

2843031836721735275017513033168065997796875

130.211813639...

21833900913142490840925787807662370512384 1656.04702793... 25297.5508014... 441706.519216...

annum*TheenergiescalculatedfromEq.(12)aredenotedasEn.TheanalyticenergiesdenotedasEnwerecalculatedforn 0,2,...,

ananan

12.Forlackofspace,theanalyticenergiesE8,E10,andE12arewrittenasadecimalnumberonly.Odd-orderperturbationenergiesequalzero.