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Asymptotics of Eigenfunctions of the Bouncing Ball Type of the Operator \( \nabla D(x)\nabla \) in a Domain Bounded by Semirigid Walls

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Abstract

We consider the problem on the semiclassical spectrum of the operator \(\nabla D(x)\nabla \) with Bessel-type degeneration on the boundary of a two-dimensional domain (semirigid walls). It is well known that the asymptotic eigenfunctions associated with Lagrangian manifolds can be constructed using a modification of the Maslov canonical operator. We obtain asymptotic eigenfunctions associated with the simplest periodic trajectories of the corresponding Hamiltonian system with reflections on the domain boundary.

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ACKNOWLEDGMENTS

The author is grateful to S.Yu. Dobrokhotov for posing this problem and assistance in research.

Funding

This work was supported by the Russian Science Foundation, project no. 16-11-10282.

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Authors and Affiliations

  1. Ishlinsky Institute for Problems in Mechanics, Russian Academy of Sciences, Moscow, 119526, Russia

    A. I. Klevin

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  1. A. I. Klevin
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Correspondence to A. I. Klevin.

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Translated by V. Potapchouck

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Klevin, A.I. Asymptotics of Eigenfunctions of the Bouncing Ball Type of the Operator \( \nabla D(x)\nabla \) in a Domain Bounded by Semirigid Walls. Diff Equat 57, 221–240 (2021). https://doi.org/10.1134/S0012266121020117

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  • DOI: https://doi.org/10.1134/S0012266121020117

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