ИСТИНА

The eigenproblem for the Laplacian inside a three-dimensional domain of revolution Ω is considered. The domain Ω is diffeomorphic to a solid torus and the Dirichlet conditions on the boundary are set. A series {Ek, uk} of asymptotic eigenvalues and eigenfunctions (quasimodes) of the whispering gallery-type is constructed (see [1]). Namely, the short-wave asymptotic eigenfunctions that are localized near the boundary (or a part of the boundary) are of interest. Reduction of the initial problem to one-dimensional ones is done using the adiabatic approximation in the form of operator separation of variables (that was discribed in [2]). The relationship between quasimodes and almost integrable classical billiards is also discussed (see Fig. 1). Lomonosov MSU with support by RSF grant 22-71-10106 http://agora.guru.ru/display.php?conf=diff-2024&page=subjects&PHPSESSID=qom0l6vkmev9e5a55jmlde1ba0