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 | References | Similar Articles | Additional Information

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 Birman-Schwinger-type 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.

**[1]**Mark S. Ashbaugh and Pavel Exner,*Lower bounds to bound state energies in bent tubes*, Phys. Lett. A**150**(1990), no. 3-4, 183–186. MR**1078396**, 10.1016/0375-9601(90)90118-8**[2]**R. Blankenbecler, M. L. Goldberger, and B. Simon,*The bound states of weakly coupled long-range one-dimensional quantum Hamiltonians*, Ann. Physics**108**(1977), no. 1, 69–78. MR**0456018****[3]**P. Duclos and P. Exner,*Curvature vs. thickness in quantum waveguides*, Czech. J. Phys.**41**(1991), 1009-1018; erratum, vol. 42, 1992, p. 344.**[4]**P. Duclos and P. Exner,*Curvature-induced bound states in quantum waveguides in two and three dimensions*, Rev. Math. Phys.**7**(1995), no. 1, 73–102. MR**1310767**, 10.1142/S0129055X95000062**[5]**D. V. Evans, M. Levitin, and D. Vassiliev,*Existence theorems for trapped modes*, J. Fluid Mech.**261**(1994), 21–31. MR**1265871**, 10.1017/S0022112094000236**[6]**Pavel Exner,*Bound states in quantum waveguides of a slowly decaying curvature*, J. Math. Phys.**34**(1993), no. 1, 23–28. MR**1198617**, 10.1063/1.530378**[7]**P. Exner and P. Šeba,*Bound states in curved quantum waveguides*, J. Math. Phys.**30**(1989), no. 11, 2574–2580. MR**1019002**, 10.1063/1.528538**[8]**P. Exner, P.\v{S}eba, and P. \v{S}\v{t}oví\v{c}ek,*On existence of a bound state in an L-shaped waveguide*, Czech. J. Phys. B**39**(1989), 1181-1191.**[9]**Tosio Kato,*Perturbation theory for linear operators*, 2nd ed., Springer-Verlag, Berlin-New York, 1976. Grundlehren der Mathematischen Wissenschaften, Band 132. MR**0407617****[10]**M. Klaus,*On the bound state of Schrödinger operators in one dimension*, Ann. Physics**108**(1977), no. 2, 288–300. MR**0503200****[11]**M. Klaus and Barry Simon,*Coupling constant thresholds in nonrelativistic quantum mechanics. I. Short-range two-body case*, Ann. Physics**130**(1980), no. 2, 251–281. MR**610664**, 10.1016/0003-4916(80)90338-3**[12]**Michael Reed and Barry Simon,*Methods of modern mathematical physics. IV. Analysis of operators*, Academic Press [Harcourt Brace Jovanovich, Publishers], New York-London, 1978. MR**0493421****[13]**W. Renger and W. Bulla,*Existence of bound states in quantum waveguides under weak conditions*, Lett. Math. Phys.**35**(1995), no. 1, 1–12. MR**1346041**, 10.1007/BF00739151**[14]**Barry Simon,*The bound state of weakly coupled Schrödinger operators in one and two dimensions*, Ann. Physics**97**(1976), no. 2, 279–288. MR**0404846**

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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

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