On the scattering vibrations in waveguides

It is known that the existence of eigenvalues of the Laplacian is an essential condition for resonance. That is, if the frequency of the time-harmonic forces belongs to some discrete set on the real axis then the solution of the corresponding non-stationary initial and boundary value problem for the wave equation with a time-harmonic right-hand side increases as t→∞. However, the absence of eigenvalues does not imply the absence of resonances as it has been shown in Wermer (1987) for the case of cylindrical waveguides with constant cross-section. In this paper a class of non-trivial solutions of the homogeneous Dirichlet´s boundary value problem for the Helmholtz equation in a two-dimensional waveguide with locally variable cross-section is studied. The feature of these solutions is that they can be not decreasing in infinity. The theorem of non-existence is obtained under fulfilment of one geometrical condition