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Trapped modes in a waveguide with a thick obstacle

Part of: Spectral theory and eigenvalue problems

Published online by Cambridge University Press:  26 February 2010

Helen Hawkins  and
Leonid Parnovski
Helen Hawkins
Affiliation:
Centre for Mathematical Analysis and Its Applications, University of Sussex, Falmer, Brighton BN1 9QH, UK. E-mail: h.l.hawkins@sussex.ac.uk
Leonid Parnovski
Affiliation:
Department of Mathematics, University of College London, Gower Street, London WC1E 6BT, UK. E-mail: Leonid@math.ucl.ac.uk.
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Extract

The problem of finding necessary and sufficient condi-tions for the existence of trapped modes in waveguides has been known since 1943. [10]. The problem is the following: consider an infinite strip M in ℝ2(or an infinite cylinder with the smooth boundary in ℝn). The spectrum of the(positive) Laplacian, with either Dirichlet or Neumann boundary conditions, acting on this strip is easily computable via the separation of variables; the spectrum is absolutely continuous and equals [v0,+∞). Here, v0 is the first threshold, i.e., eigenvalue of the cross-section of the cylinder (so v0 = 0 in the case of Neumann conditions). Let us now consider the domain (the waveguide) which is a smooth compact perturbation of M (for example, weinsert an obstacle inside M). The essential spectrum of the Laplacian acting on still equals [v0, +ℝ), but there may be additional eigenvalues, which are often called trapped modes; the number of these trapped modes can be quite large, see examples in [11] and [8].

MSC classification

Secondary: 35P15: Estimation of eigenvalues, upper and lower bounds
Type
Research Article
Information
Mathematika , Volume 51 , Issue 1-2 , December 2004 , pp. 171 - 186
Copyright
Copyright © University College London 2004

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References

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