New Version of the Rayleigh–Schrdinger Perturbation Theory(10)

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

TABLEVI_____________________________________________________________________________________________

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

3.65

2

0.1629972

4

0.403834

6

2.117584

81012141618202224262830

15.76930 150.1364 1730.805 23374.8 0.361673 106 0.63081 107 0.1224862 109 0.262120 1010 0.613108 1011 0.155649 1013 0.426302 1014 0.1253085 1016

numEn

anEn

73

3.6520

1973875 0.162997288821...12109864

448842824539509375

0.403834899438...

529018839425166255132197044609375

2.11758446234...

249821836546340968688607521245696

733184603275850651131746100573669230477374181640625

15.7693083287...

150.136468953... 1730.80567546...

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

anan

12.Forlackofspace,theanalyticenergiesE10andE12arewrittenasadecimalnumberonly.Odd-orderperturbationenergiesequalzero.

TABLEVII____________________________________________________________________________________________

PerturbationenergiesEnforthethirdexcited[1,1]stateoftheHe´non–HeilesHamiltonian(31,32);E0isthe

*exactzero-orderenergy.n0

4.95

2

1.845463

4

8.85123

68

1012141618202224262830

78.03980 919.9970 13100.3 214537.0 0.3931028 107 0.79242 108 0.1737966 1010 0.411554 1011 0.10463977 1013 0.28444762 1014 0.823863 1015 0.2535079 1017 0.826589 1018

numEn

anEn

99

4.9520

201134875

1.84546411458...

21515087825404789503125

8.85123561303...

2430743996206636089408

694186987069655474213702464188191838828125

78.0398115982...

919.997135266... 13100.3815465... 214537.066115...

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

anan

12.Forlackofspace,theanalyticenergiesE10andE12arewrittenasadecimalnumberonly.Odd-orderperturbationenergiesequalzero.