Abstract

In this talk, we are interested in the spectrum of the Dirichlet Laplacian in thin domains. First, we consider broken strips of thickness ε>0 small with an angle α. We construct an asymptotic expansion of the first eigenvalues and corresponding eigenfunctions when ε tends to zero. By studying the dependence of this expansion with respect to α, we highlight a curious phenomenon of “diving eigenvalues”: when the strip is more and more broken, at certain critical angles, an eigenvalue moves down rapidly below the pack of other eigenvalues. By exploiting this analysis, in a second step, we present another intriguing phenomenon of “spectral breathing” in quasi 1D periodic waveguides: when changing continuously the geometry around certain configurations, spectral brands, which are generically very short due to the Dirichlet boundary conditions, can grow to a considerable length. This is a joint work with S.A. Nazarov.