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A lower bound to the spectral threshold in curved tubes. (English) Zbl 1330.35360

Summary: We consider the Laplacian in curved tubes of arbitrary cross-section rotating together with the Frenet frame along curves in Euclidean spaces of arbitrary dimension, subject to Dirichlet boundary conditions on the cylindrical surface and Neumann conditions at the ends of the tube. We prove that the spectral threshold of the Laplacian is estimated from below by the lowest eigenvalue of the Dirichlet Laplacian in a torus determined by the geometry of the tube.

MSC:

35Q40 PDEs in connection with quantum mechanics
81Q10 Selfadjoint operator theory in quantum theory, including spectral analysis
47N50 Applications of operator theory in the physical sciences
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