Some properties of WKB series

Abstract

We investigate some properties of the WKB series for arbitrary analytic potentials and then specifically for potentials $x^N$ ($N$ even), where more explicit formulae for the WKB terms are derived. Our main new results are: (i) We find the explicit functional form for the general WKB terms ${\sigma }_k^{\prime }$, where one has only to solve a general recursion relation for the rational coefficients. (ii) We give a systematic algorithm for a dramatic simplification of the integrated WKB terms $\oint {\sigma }_k^{\prime }dx$ that enter the energy eigenvalue equation. (iii) We derive almost explicit formulae for the WKB terms for the energy eigenvalues of the homogeneous power law potentials $V(x)=x^N$, where $N$ is even. In particular, we obtain effective algorithms to compute and reduce the terms of these series.

18 pages, submitted to Journal of Physics A: Mathematical and General

References in corpus (1)

• On WKB Series for the Radial Kepler Problem

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