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Curvature-induced bound states in quantum waveguides in two and three dimensions. (English) Zbl 0837.35037

Summary: Dirichlet Laplacian on curved tubes of a constant cross section in two and three dimensions is investigated. It is shown that if the tube is non-straight and its curvature vanishes asymptotically, there is always a bound state below the bottom of the essential spectrum. An upper bound to the number of these bound states in this tubes is derived. Furthermore, if the tube is only slightly bent, there is just one bound state, we derive its behaviour with respect to the bending angle. Finally, perturbation theory of these eigenvalues in any thin tube with respect to the tube radius is constructed and some open questions are formulated.

MSC:

35J05 Laplace operator, Helmholtz equation (reduced wave equation), Poisson equation
81V10 Electromagnetic interaction; quantum electrodynamics
35P05 General topics in linear spectral theory for PDEs
35J10 Schrödinger operator, Schrödinger equation
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