Influence of bounded states in the Neumann Laplacian in a thin waveguide

Abstract

Let $-\Delta _\Omega ^N$ be the Neumann Laplacian operator restricted to a twisted waveguide $\Omega $. Our first goal is to find the effective operator when $\Omega $ is ``squeezed.'' However, since, in this process, there are divergent eigenvalues, we consider $-\Delta _\Omega ^N$ acting in specific subspaces of the initial Hilbert space. The strategy is interesting since we find different effective operators in each situation. In the case where $\Omega $ is periodic and sufficiently thin, we also obtain information regarding the absolutely continuous spectrum of $-\Delta _\Omega ^N$ (restricted to such subspaces) and the existence and location of band gaps in its structure.