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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. [ABSTRACT FROM AUTHOR]

We establish that the principal eigenfunction of the Dirichlet problem in a domain with a thin heavy edging admits localization near the corner point of opening angle α > π. The edging amounts to a boundary strip of small width ɛ with the density function ɛ, m > 0, while it is O(1) in the remaining part of the domain. We derive the result by analyzing the essential and discrete spectra of an auxiliary problem in an infinite angle without the small parameter. We state several open questions about the structure of spectra of both problems. [ABSTRACT FROM AUTHOR]