ИСТИНА

The 2D Schrodinger equation with variable potential in the narrow domain diffeomorphic to the wedge with the Dirichlet boundary condition is considered. 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 (see [V. I. Arnold, V. V. Kozlov and A. I. Neishtadt, 2006]). The quantum adiabatic approximation (in the form of operator separation of variables [S. Yu. Dobrokhotov, 1983]) is used to construct the asymptotic eigenfunctions (quasimodes) of the Schrodinger operator. The relation between classical averaging and constructed quasimodes is discussed and the behavior of quasimodes in the neighborhood of the cusp is studied. Also the relation between Bessel and Airy functions that follows from different representations of asymptotics near the cusp is obtained. The talk is done based on results obtained together with S. Yu. Dobrokhotov, A. I. Neishtadt and S. B. Shlosman [S. Yu. Dobrokhotov, D. S. Minenkov, A. I. Neishtadt and S. B. Shlosman, 2019] and is supported by RSF grant 24-11-00213. https://indico.eimi.ru/event/1893/attachments/432/912/Birman2025_-_Proceedings.pdf