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Эта публикация цитируется в 2 научных статьях (всего в 2 статьях)
Classical and Quantum Dynamics of a Particle in a Narrow Angle
Sergei Yu. Dobrokhotovab, Dmitrii S. Minenkovba, Anatoly I. Neishtadtcd, Semen B. Shlosmanefg a Ishlinsky Institute for Problems in Mechanics of the Russian Academy of Sciences (IPMech RAS),
prosp. Vernadskogo 101, Moscow, 119526 Russia
b Moscow Institute of Physics and Technology,
Institutskii per. 9, Dolgoprudnyi, 141701 Russia
c Space Research Institute,
Profsoyuznaya ul. 84/32, Moscow, 117997 Russia
d Loughborough University,
Epinal Way, Loughborough Leicestershire, UK
e Aix Marseille Univ, Universite de Toulon,
CNRS, CPT, Marseille, France
f Institute of the Information Transmission Problems, RAS,
Bolshoy Karetny per. 19, Moscow, 127051 Russia
g Skolkovo Institute of Science and Technology,
Nobel ul. 3, Moscow, 121205 Russia
Аннотация:
We consider the 2D Schrödinger 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 (quasi-modes) of the Schröodinger operator. We discuss the relation between classical averaging and constructed quasi-modes. The behavior of quasi-modes 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.
Ключевые слова:
potential well, stationary Schrödinger equation, KAM theory, operator separation of variables, semiclassical asymptotics, Airy function, Bessel function.
Поступила в редакцию: 19.07.2019 Принята в печать: 18.10.2019
Образец цитирования:
Sergei Yu. Dobrokhotov, Dmitrii S. Minenkov, Anatoly I. Neishtadt, Semen B. Shlosman, “Classical and Quantum Dynamics of a Particle in a Narrow Angle”, Regul. Chaotic Dyn., 24:6 (2019), 704–716
Образцы ссылок на эту страницу:
https://www.mathnet.ru/rus/rcd1034 https://www.mathnet.ru/rus/rcd/v24/i6/p704
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