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Weakly Coupled Bound States In Quantum Waveguides

By Bulla Gesztesy, W. Bulla, F. Gesztesy, W. Renger, B. Simon and Ae R


. We study the eigenvalue spectrum of Dirichlet Laplacians which model quantum waveguides associated with tubular regions outside of a bounded domain. Intuitively, our principal new result in two dimensions asserts that any domain\Omega obtained by adding an arbitrarily small "bump" to the tube\Omega 0 = R \Theta (0; 1) (i.e., \Omega %\Omega 0 ,\Omega ae R 2 open and connected,\Omega =\Omega 0 outside a bounded region) produces at least one positive eigenvalue below the essential spectrum [ß 2 ; 1) of the Dirichlet Laplacian \Gamma\Delta D \Omega . For j\Omega n\Omega 0 j sufficiently small (j : j abbreviating Lebesgue measure), we prove uniqueness of the ground state E\Omega of \Gamma\Delta D\Omega and derive the "weak coupling" result E\Omega = ß 2 \Gamma ß 4 j\Omega n\Omega 0 j 2 +O(j\Omega n\Omega 0 j 3 ). As a corollary of these results we obtain the following surprising fact: Starting from the tube\Omega 0 with Dirichlet boundary conditions at @\Om..

Year: 1995
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