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Resurgent methods in semi-classical asymptotics. (English) Zbl 0977.34053

Here, the authors discuss WKB expansions for one-dimensional Schrödinger equations, especially the problem of quadratic confluence, i.e., confluence of two simple turning points, from the viewpoint of resurgence theory.
After giving an overview on the theory of resurgent functions together with considerations concerning their dependence on parameters, the authors re-explain their previous results with H. Dillinger in generic situations (e.g., under the hypothesis that all turning points are simple [Ann. Inst. Fourier, 43, No. 1, 163-199 (1993; Zbl 0766.34032)]. Then, using these previous results, they discuss the quadratic confluence: They give an expression for the connection formula in terms of the Euler gamma function and an expression for wave functions in terms of Weber parabolic cylinder functions. They also explain how to perform explicit computations in all orders on what they call the “rescaled” Schrödinger equation.
Inspired by the results and ideas of Ecalle and Voros, the authors have been working to establish the resurgent theoretical treatment of one-dimensional WKB expansions in a mathematically rigorous manner. The present paper is one of the outcomes of their efforts. This is, on one hand, a research paper on the quadratic confluence and, on the other hand, a good expository article introducing the theory of resurgent functions.

MSC:

34E20 Singular perturbations, turning point theory, WKB methods for ordinary differential equations
81Q20 Semiclassical techniques, including WKB and Maslov methods applied to problems in quantum theory
34M37 Resurgence phenomena (MSC2000)

Citations:

Zbl 0766.34032
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References:

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