1 research outputs found
Classical and quantum dynamics of a particle in a narrow angle
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