Semen Shlosman
Publications:
Dobrokhotov S. Y., Minenkov D. S., Neishtadt A. I., Shlosman S. B.
Classical and Quantum Dynamics of a Particle in a Narrow Angle
2019, vol. 24, no. 6, pp. 704716
Abstract
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 (quasimodes) of the Schr¨odinger 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.
