ИСТИНА

The eigenproblem for the Laplacian inside a three-dimensional domain of revolution is considered. The domain is diffeomorphic to a solid torus and the Dirichlet conditions on the boundary are set. A series of asymptotic eigenvalues and eigenfunctions (quasimodes) of the whispering gallery-type is constructed. Namely, the short-wave asymptotic eigenfunctions that are localized near the boundary (or a part of the boundary) are of interest. Such quasimodes are related to almost integrable billiard trajectories, that are localized near the boundary. Reduction of the quantum problem inside the solid torus to one-dimensional ones is done using the adiabatic approximation in the form of operator separation of variables. This procedure is closely related to averaging of mentioned billiard trajectories, particularly, the averaged classical Hamiltonian is a main symbol of the reduced quantum operator. RSF grant 22-71-10106. http://isqt.tilda.ws/