On the interior transmission eigenvalue problem

We consider the transmission eigenvalue problem corresponding to the scattering problem for anisotropic media for both the scalar Helmholtz equation and Maxwell's equations in the case when the contrast in the scattering media occurs in two independent functions. We prove the existence of an infinite discrete set of transmission eigenvalues provided that the two contrasts are of opposite signs. In this case we provide bounds for the first transmission eigenvalue in terms of the ratio of refractive indices. In the case of the same sign contrasts for the scalar case we show the existence of a finite number of transmission eigenvalues under restrictive assumptions on the strength of the scattering media.