Weakly coupled bound states in quantum waveguides
Authors:
W. Bulla, F. Gesztesy, W. Renger and B. Simon
Journal:
Proc. Amer. Math. Soc. 125 (1997), 14871495
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 obtained by adding an arbitrarily small ``bump'' to the tube (i.e., , open and connected, outside a bounded region) produces at least one positive eigenvalue below the essential spectrum of the Dirichlet Laplacian . For sufficiently small ( abbreviating Lebesgue measure), we prove uniqueness of the ground state of and derive the ``weak coupling'' result using a BirmanSchwingertype analysis. As a corollary of these results we obtain the following surprising fact: Starting from the tube with Dirichlet boundary conditions at , replace the Dirichlet condition by a Neumann boundary condition on an arbitrarily small segment , , of . If denotes the resulting Laplace operator in , then has a discrete eigenvalue in no matter how small is.
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Additional Information
W. Bulla
Affiliation:
Institute for Theoretical Physics, Technical University of Graz, A8010 Graz, Austria
Email:
bulla@itp.tugraz.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/S000299399703726X
PII:
S 00029939(97)03726X
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. DMS9401491. The Government has certain rights in this material.
Communicated by:
Palle E. T. Jorgensen
Article copyright:
© Copyright 1997
by the authors
