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Influence of bounded states in the Neumann Laplacian in a thin waveguide. (English) Zbl 1404.35131
Summary: 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.
35J10 Schrödinger operator, Schrödinger equation
35P99 Spectral theory and eigenvalue problems for partial differential equations
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