Billiards with semi-rigid walls, asymptotic eigenfunctions of the 2D operator $\nabla D(x)\nabla$, and trapped coastal waves

We construct asymptotic eigenfunctions of the two-dimensional operator $\widehat L=\nabla D(x)\nabla$ in a domain $\Omega$ with coefficient $D(x)$ degenerating on the boundary $\partial\Omega$. These eigenfunctions are associated with Liouville tori of integrable geodesic flows with a metric degenerating on $\partial\Omega$. Such geodesic flows can be called “billiards with semi-rigid walls”.
The talk is based on joint work with A.Yu.Anikin, S.Yu.Dobrokhotov, and A.V.Tsvetkova. The research was supported by the Russian Science Foundation under grant no. 16-11-10282.