Hawkins, H; Parnovski, L; (2004) Trapped modes in a waveguide with a thick obstacle. MATHEMATIKA , 51 (101-02) 171 - 186. 10.1112/S0025579300015606.

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Abstract

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 xs211D2(or an infinite cylinder with the smooth boundary in xs211Dn). 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 S0025579300015606_inline1 (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 S0025579300015606_inline1 still equals [v0, +xs211D), but there may be additional eigenvalues, which are often called trapped modes; the number of these trapped modes can be quite large.

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