Dispersive hyperasymptotics and the anharmonic oscillator. (English) Zbl 1042.81018
Summary: Hyperasymptotic summation of steepest-descent asymptotic expansions of integrals is extended to functions that satisfy a dispersion relation. We apply the method to energy eigenvalues of the anharmonic oscillator, for which there is no known integral representation, but for which there is a dispersion relation. Hyperasymptotic summation exploits the rich analytic structure underlying the asymptotics and is a practical alternative to Borel summation of the Rayleigh-Schrödinger perturbation series.
MSC:
| 81Q15 | Perturbation theories for operators and differential equations in quantum theory |
| 34E05 | Asymptotic expansions of solutions to ordinary differential equations |
| 34E10 | Perturbations, asymptotics of solutions to ordinary differential equations |
| 34E18 | Methods of nonstandard analysis for ordinary differential equations |
| 81U20 | \(S\)-matrix theory, etc. in quantum theory |
