Different types of localization for eigenfunctions of scalar mixed boundary value problems in thin polyhedra

Abstract

We construct asymptotics for the eigenvalues and eigenfunctions of the Laplace operator in a thin polyhedron with
parallel closely spaced bases and skewed narrow lateral faces. On the bases we impose the Dirichlet conditions, while on the lateral faces the Dirichlet or Neumann conditions are imposed. Their distribution over the faces, as well as the slope of the latter, significantly affect the behavior of eigenfunctions when the domain becomes thinner. We find situations, in which the eigenfunctions are distributed along the entire polyhedron and localized near its lateral faces or vertices. The results are based on the analysis of the spectrum (cut--off point, isolated eigenvalues, threshold resonances, etc.) of auxiliary problems in a half--strip and a quarter of a layer with skewed end and lateral sides, respectively. We formulate open questions concerning both spectral and asymptotic analysis.