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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.
##### MSC:
 35J10 Schrödinger operator, Schrödinger equation 35P99 Spectral theory and eigenvalue problems for partial differential equations
##### Keywords:
Neumann Laplacian operator; twisted waveguide; eigenvalues
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##### References:
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