Eigenvalues and scattering frequencies for domains with narrow appendixes and tubes

Abstract

The eigenvalue problem of the Dirichlet Laplacian in a singularly perturbed region, which is described as a bounded domain with a thin appendix is considered. The expansion of eigenvalues in the power series with respect to a small parameter (radius of the cross-section of the appendix) is constructed. This result is extended to the Helmholtz resonator with the finite thickness of a shell.