Inverse scattering for a penetrable cavity and the transmission eigenvalue problem

Abstract

In this thesis, we consider the scattering of point sources inside a cavity surrounded by an inhomogeneous medium and its inverse problem of determining the boundary of the cavity from measurements of the scattered field inside the cavity. We apply the linear sampling method and factorization method to numerically reconstruct the boundary of the cavity. We prove that the linear sampling method works when the wave number is not an exterior transmission eigenvalue. We prove that the exterior transmission eigenvalues form a discrete set. We then consider both the exterior transmission eigenvalue problem and the interior transmission eigenvalue problem for a spherically stratified media and study the inverse spectral problem for the exterior transmission eigenvalue problem. Finally we consider the interior transmission eigenvalue problem for Maxwell's equations corresponding to non-magnetic inhomogeneities with contrast in electric permittivity that changes sign inside its support. We prove that the set of transmission eigenvalues is nonempty discrete, infinite and without finite accumulation points.