Anharmonic Oscillator. II. A Study of Perturbation Theory in Large Order

This paper is concerned with the nature of perturbation theory in very high order. Specifically, we study the Rayleigh-Schr\"odinger expansion of the energy eigenvalues of the anharmonic oscillator. We have developed two independent mathematical techniques (WKB analysis and difference-equation methods) for determining the large-n behavior of AₙK, the nth Rayleigh-Schr\"odinger coefficient for the Kth energy level. We are not concerned here with placing bounds on the growth of AₙK as n, the order of perturbation theory, gets large. Rather, we consider the more delicate problem of determining the precise asymptotic behavior of AₙK as n→∞ for both the Wick-ordered and non-Wick-ordered oscillators. Our results are in exact agreement with numerical fits obtained from computer studies of the anharmonic oscillator to order 150 in perturbation theory.