Numerological analysis of the WKB approximation in large order

Abstract

We show how to slove the one-dimensional two-turning-point eigenvalue problem for analytic potentials to all orders in the WKB approximation. We use this method to compute the eigenvalues of the ${x}^{N}$ ($N$ even) potential to twelfth order. Numerical results for the ${x}^{4}$ potential are accurate to 1 part in ${10}^{15}$ for the tenth eigenvalue. For the ${\ensuremath{\nu}}_{0}{cosh}^{\ensuremath{-}2}x$ potential the WKB series reduces to a geometric series which may be summed to give the exact answer. Finally, we report on the results of numerological experiments on the structure of the WKB series. The simplicity of our results leads us to conjecture (weakly) that it may be possible to find a formula for the terms of the WKB series for arbitrary analytic potentials.