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Exact semiclassical expansions for one-dimensional quantum oscillators. (English) Zbl 0896.34051

The time independent one-dimensional Schrödinger equation is considered. WKB expansions encoding exact (not only approximate) solutions of the Schrödinger equation are studied. Rules for computing the Stokes automorphism and the connection isomorphisms are given. Generic values of energy for which all turning points are simple are dealt with. The connection isomorphisms are given by combinations of analytic continuations along suitable paths of the complex \(q\) plane and they are described explicitly by simple pictograms. The pictograms are extended to the case, when there are double turning points. The corresponding WKB expansions then involve special prefactors. The problem of solving the quantization condition for bound states or resonances with respect to the energy parameter is examined. Expressions for the energy levels yielding rigorous justifications of such results as the Zinn-Justin expansions are obtained.
Reviewer: V.Burjan (Praha)

MSC:

34E20 Singular perturbations, turning point theory, WKB methods for ordinary differential equations
34L40 Particular ordinary differential operators (Dirac, one-dimensional Schrödinger, etc.)
81Q20 Semiclassical techniques, including WKB and Maslov methods applied to problems in quantum theory
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