The Interior Transmission Eigenvalue Problem

We consider the inverse problem of determining the spherically symmetric index of refraction $n(r)$ from a knowledge of the corresponding transmission eigenvalues (which can be determined from field pattern of the scattered wave). We also show that for constant index of refraction $n(r)=n$, the smallest transmission eigenvalue suffices to determine n, complex eigenvalues exist for n sufficiently small and, for homogeneous media of general shape, determine a region in the complex plane where complex eigenvalues must lie.