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Published Online: 04 June 1998
Accepted: June 1997

Exact semiclassical expansions for one-dimensional quantum oscillators

Journal of Mathematical Physics 38, 6126 (1997); https://doi.org/10.1063/1.532206
Eric Delabaere
  • UMR CNRS J. A. Dieudonné No. 6621, University of Nice, 06108 Nice Cedex 2, France
    • Hervé Dillinger and Frédéric Pham
  • University of Nice, Department of Maths, UMR CNRS J.A. Dieudonné No. 6621, 06108 Nice Cedex 2, France
    • more...
    Abstract
    A set of rules is given for dealing with WKB expansions in the one-dimensional analytic case, whereby such expansions are not considered as approximations but as exact encodings of wave functions, thus allowing for analytic continuation with respect to whichever parameters the potential function depends on, with an exact control of small exponential effects. These rules, which include also the case when there are double turning points, are illustrated on various examples, and applied to the study of bound state or resonance spectra. In the case of simple oscillators, it is thus shown that the Rayleigh–Schrödinger series is Borel resummable, yielding the exact energy levels. In the case of the symmetrical anharmonic oscillator, one gets a simple and rigorous justification of the Zinn-Justin quantization condition, and of its solution in terms of “multi-instanton expansions.”

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