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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

   

 

Weakly coupled bound states
in quantum waveguides


Authors: W. Bulla, F. Gesztesy, W. Renger and B. Simon
Journal: Proc. Amer. Math. Soc. 125 (1997), 1487-1495
MSC (1991): Primary 81Q10, 35P15; Secondary 47A10, 35J10
MathSciNet review: 1371117
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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 $\Omega _{0}=\mathbb {R}\times (0,1)$ (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 $(a,b)\times \{1\}$, $a<b$, of $\partial \Omega _{0}$. 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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Additional Information

W. Bulla
Affiliation: Institute for Theoretical Physics, Technical University of Graz, A-8010 Graz, Austria
Email: bulla@itp.tu-graz.ac.at

F. Gesztesy
Affiliation: Department of Mathematics, University of Missouri, Columbia, Missouri 65211
Email: fritz@math.missouri.edu

W. Renger
Affiliation: Department of Mathematics, University of Missouri, Columbia, Missouri 65211
Email: walter@mumathnx3.cs.missouri.edu

B. Simon
Affiliation: Division of Physics, Mathematics, and Astronomy, California Institute of Technology, Pasadena, California 91125

DOI: http://dx.doi.org/10.1090/S0002-9939-97-03726-X
PII: S 0002-9939(97)03726-X
Keywords: Dirichlet Laplacians, waveguides, ground states
Received by editor(s): November 13, 1995
Additional Notes: This material is based upon work supported by the National Science Foundation under Grant No. DMS-9401491. The Government has certain rights in this material.
Communicated by: Palle E. T. Jorgensen
Article copyright: © Copyright 1997 by the authors