Nonlinear Phenomena in Complex Systems, 2000, Vol.3, No.3, pp.214--219

NONLINEAR PHENOMENA IN COMPLEX SYSTEMS
An Interdisciplinary Journal

2000, Volume 3, Number 3, pp.214--219

On WKB Series for the Radial Kepler Problem.
Valery Romanovski and Marko Robnik

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 Schrödinger 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.
Key words: radial Kepler problem, WKB expansion, Schrödinger eigenvalue problem.

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