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# Classical and quantum dynamics of a particle in a narrow angle

journal contribution
posted on 08.10.2019, 09:10 by S Dobrokhotov, D Minenkov, 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.

• Science

## Department

• Mathematical Sciences

## Published in

Regular and Chaotic Dynamics

24

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

03/10/2019

2019-12-10

2019

1468-4845

en

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