Infinite Kirchhoff plate on a compact elastic foundation may have arbitrary small eigenvalue

Abstract

An inhomogeneous Kirhhoff plate composed from semi-infinite strip-waveguide and a compaсt resonator which is in contact with the Winkler foundation of small compliance, is considered. It is shown that for any 0, it is possible to find the compliance coefficient O(2) such that the described plate possesses the eigenvalue 4embedded into continuous spectrum. This result is quite surprising because in an acoustic waveguide (the spectral Neumann problem for the Laplace operator) a small eigenvalue does not exist for any unsubstantial perturbation. A reason of this dissension is explained as well.