Response behavior of embedded eigenvalues under changes of waveguide filling

In the context of perturbation theory it is demonstrated that embedded eigenvalues of a spectral problem for a waveguide filled with inhomogeneous substance change into complex resonances under small perturbation of material filling. For this purpose it has been proved that already in the first order of perturbation theory eigenfunctions do not belong to the space L$^2$, although the eigenvalues corresponding to them remain real in the same order of smallness.