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 ℝ 2 (or an infinite cylinder with the smooth boundary in ℝ n ). 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 [ v 0 ,+∞). Here, v 0 is the first threshold, i.e. , eigenvalue of the cross-section of the cylinder (so v 0 = 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 [ v 0 , +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].