Abstract
We construct asymptotic eigenfunctions for the two-dimensional Schrödinger operator with a potential in the form of a well that is mirror-symmetric with respect to a line. These functions correspond to librations on this line between two focal points. According to the Maslov complex germ theory, the asymptotic eigenfunctions in the direction transverse to the line with respect to which the well is symmetric have the form of the appropriate Hermite-Gauss mode. We obtain a global Airy-function representation for the asymptotic eigenfunctions in the longitudinal direction.
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The author thanks S. Yu. Dobrokhotov for formulating the problem and assisting in this work.
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This research is supported by a grant from the Russian Science Foundation (Project No. 16-11-10282).
Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 199, No. 3, pp. 429–444, June, 2019.
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Klevin, A.I. Asymptotic Eigenfunctions of the “Bouncing Ball” Type for the Two-Dimensional Schrödinger Operator with a Symmetric Potential. Theor Math Phys 199, 849–863 (2019). https://doi.org/10.1134/S0040577919060060
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DOI: https://doi.org/10.1134/S0040577919060060