 95422 W. Bulla, F. Gesztesy, W. Renger, B. Simon
 Weakly Coupled Bound States in Quantum Waveguides
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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 ba>0 is.
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