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Classical and quantum dynamics of a particle in a narrow angle

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journal contribution
posted on 08.10.2019 by S Dobrokhotov, D Minenkov, Anatoly Neishtadt, S Shlosman
We consider the 2D Schr¨odinger equation with variable potential in the narrow domain diffeomorphic to the wedge with the Dirichlet boundary condition. The corresponding classical problem is the billiard in this domain. In general, the corresponding dynamical system is not integrable. The small angle is a small parameter which allows one to make the averaging and reduce the classical dynamical system to an integrable one modulo exponential small correction. We use the quantum adiabatic approximation (operator separation of variables) to construct the asymptotic eigenfunctions (quasimodes) of the Schrodinger operator. We discuss the relation between classical averaging and constructed quasimodes. The behavior of quasimodes in the neighborhood of the cusp is studied. We also discuss the relation between Bessel and Airy functions that follows from different representations of asymptotics near the cusp.

Funding

RFBR–CNRS project 17-51-150006.

History

School

  • Science

Department

  • Mathematical Sciences

Published in

Regular and Chaotic Dynamics

Volume

24

Pages

704–716

Publisher

MAIK Nauka/Interperiodica

Version

AM (Accepted Manuscript)

Publisher statement

This paper was accepted for publication in the journal Regular and Chaotic Dynamics and the definitive published version is available at https://doi.org/10.1134/S156035471906008X

Acceptance date

03/10/2019

Publication date

2019-12-10

Copyright date

2019

ISSN

1468-4845

Language

en

Depositor

Prof Anatoly Neishtadt

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