95-422 W. Bulla, F. Gesztesy, W. Renger, B. Simon
Weakly Coupled Bound States in Quantum Waveguides (29K, amstex) Sep 12, 95
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Abstract. 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 x (0,1) produces at least one positive eigenvalue below the essential spectrum [\pi^2,\infty) of the Dirichlet Laplacian corresponding to \Omega. For |\Omega\ \Omega_0| sufficiently small, we prove uniqueness of the associated ground state E_{\Omega} and derive the weak coupling result E_{\Omega} = \pi^2 - \pi^4|\Omega\ \Omega_0|^2 + O(|\Omega\ \Omega_0|^3). As a corollary of these results we obtain the following surprising fact: Starting from the tube \Omega_0 with Dirichlet boundary conditions on its boundary, replace the Dirichlet condition by a Neumann boundary condition on an arbitrarily small segment (a,b) x {1}, a<b. If H(a,b) denotes the resulting Laplace operator in L^2(\Omega_0), then H(a,b) has a discrete eigenvalue in [\pi^2/4,\pi^2) no matter how small |b-a|>0 is.

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