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 ${A}_{n}^{K}$, the nth Rayleigh-Schr\"odinger 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 $n\ensuremath{\rightarrow}\ensuremath{\infty}$ 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.