Localization of eigenfunctions in the Dirichlet beaker

Abstract

AbstractWe construct the asymptotics of the eigenpairs of the Dirichlet problem for the Laplace operator in a thin‐walled beaker and prove the localization effect for the functions near the bottom edge, a smooth closed contour, of the beaker. The main asymptotic terms are described by the eigenpairs of an ordinary differential equation on the edge and by the single eigenvalue belonging to the discrete spectrum of the Dirichlet Laplacian in an ‐shaped infinite waveguide. The corresponding eigenfunctions are shown to decay exponentially at some distance from the edge. Also, we find the asymptotics of eigenvalue sequences generated by planar Dirichlet problems on the bottom and walls of the limit beaker of zero thickness. Open questions related to other sequences of eigenvalues are discussed.