On the existence of transmission eigenvalues

The investigation of the far field operator and the Factorization Method ininverse scattering theory leads naturally to the study of corresponding interiortransmission eigenvalue problems. In contrast to the classical Dirichlet- orNeumann eigenvalue problem for -Δ in bounded domains these interiortransmiision eigenvalue problem fail to be selfadjoint. In general, existence ofeigenvalues is an open problem. In this paper we prove existence of eigenvaluesfor the scalar Helmholtz equation (isotropic and anisotropic cases) andMaxwell's equations under the condition that the contrast of the scatteringmedium is large enough.