Computation of interior transmission eigenvalues from far field data
It is well-known that the interior eigenvalues of the Laplacian in a bounded domain share connections to scattering problems posed in the exterior of this domain. For instance, certain boundary integral equations for exterior scattering problems fail at interior resonances.
Similar connections also exist for inverse scattering problems: If, e.g., zero is an eigenvalue of the far field operator, then the squared wave number is an interior eigenvalue. Despite it is in general wrong that interior eigenvalues correspond to zero being an eigenvalue of the far field operator, one can prove a pretty direct characterization of interior eigenvalues via the behaviour of the phases of the eigenvalues of the far field operator.
In this talk, we present similar analytic characterizations of interior transmission eigenvalues for impenetrable scatterers. The analytic results include anisotropic scatterers and/or electromagnetic scattering, and they can further be exploited numerically.