The localization for eigenfunctions of the Dirichlet problem in thin polyhedra near the vertices

Under some geometric assumptions, we show that eigenfunctions of the Dirichlet problem for the Laplace operator in an $n$-dimensional thin polyhedron localize near one of its vertices. We construct and justify asymptotics for the eigenvalues and eigenfunctions. For waveguides, which are thin layers between periodic polyhedral surfaces, we establish the presence of gaps and find asymptotics for their geometric characteristics.