On WKB Series for the Radial Kepler Problem

TL;DR

This paper derives an exact all-orders WKB expansion for the radial Kepler problem, providing a rigorous and convergent series that yields the precise energy spectrum and clarifies longstanding controversies.

Contribution

It introduces a method to compute the complete WKB series for the radial Kepler problem, resolving the Langer correction controversy and offering insights into convergent series in quantum mechanics.

Findings

Exact energy spectrum obtained from the WKB series.

All orders of the WKB series are computed and shown to converge.

Clarification of the Langer correction controversy.

Abstract

We obtain the rigorous WKB expansion to all orders for the radial Kepler problem, using the residue calculus in evaluating the WKB quantization condition in terms of a complex contour integral in the complexified coordinate plane. The procedure yields the exact energy spectrum of this Schr\"odinger eigenvalue problem and thus resolves the controversies around the so-called "Langer correction". The problem is nontrivial also because there are only a few systems for which all orders of the WKB series can be calculated, yielding a convergent series whose sum is equal to the exact result, and thus sheds new light to similar and more difficult problems.