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-Schrödinger 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 $A_n^K$, the nth Rayleigh-Schrödinger coefficient for the Kth energy level. We are not concerned here with placing bounds on the growth of $A_n^K$ as $n$, the order of perturbation theory, gets large. Rather, we consider the more delicate problem of determining the precise asymptotic behavior of $A_n^K$ as $n\to \infty$ 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.