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Weakly coupled bound states in quantum waveguides

Authors:	W Bulla F Gesztesy W Renger B Simon

Institution:	Institute for Theoretical Physics, Technical University of Graz, A-8010 Graz, Austria ; Department of Mathematics, University of Missouri, Columbia, Missouri 65211 ; Department of Mathematics, University of Missouri, Columbia, Missouri 65211 ; Division of Physics, Mathematics, and Astronomy, California Institute of Technology, Pasadena, California 91125

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 $\Omega _{0}=\mathbb {R}\times (0,1)$ (i.e., $\Omega \supsetneqq \Omega _{0}$ , $\Omega \subset \mathbb {R}^{2}$ open and connected, $\Omega =\Omega _{0}$ outside a bounded region) produces at least one positive eigenvalue below the essential spectrum $\pi ^{2},\infty )$ of the Dirichlet Laplacian $-\Delta ^{D}_{\Omega }$ . For $\|\Omega \backslash \Omega _{0}\|$ sufficiently small ( $\|\,.\,\|$ abbreviating Lebesgue measure), we prove uniqueness of the ground state $E_{\Omega }$ of $-\Delta ^{D}_{\Omega }$ and derive the ``weak coupling' result $E_{\Omega }=\pi ^{2}-\pi ^{4}\|\Omega \backslash \Omega _{0}\|^{2} +O(\|\Omega \backslash \Omega _{0}\|^{3})$ using a Birman-Schwinger-type analysis. As a corollary of these results we obtain the following surprising fact: Starting from the tube $\Omega _{0}$ with Dirichlet boundary conditions at $\partial \Omega _{0}$ , replace the Dirichlet condition by a Neumann boundary condition on an arbitrarily small segment $(a,b)\times \{1\}$ , , of $\partial \Omega _{0}$ . If denotes the resulting Laplace operator in $L^{2}(\Omega _{0})$ , then has a discrete eigenvalue in $\pi ^{2} /4,\pi ^{2})$ no matter how small is.

Keywords:	Dirichlet Laplacians waveguides ground states

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