Inverse Spectral Problems for Transmission Eigenvalues

Abstract

We previously encountered transmission eigenvalues and their role in inverse scattering theory in Chap. 6. We now return to this topic and consider the inverse spectral problem for transmission eigenvalues in the simplest possible case, i.e., when the inhomogeneous medium is an isotropic spherically stratified medium in ℝ3 and the eigenfunctions corresponding to the transmission eigenvalues are spherically symmetric. In this case the inverse spectral problem for transmission eigenvalues reduces to a problem in ordinary differential equations analogous to the inverse Sturm–Liouville problem (cf. [98, 142]) with the important distinction that the spectral problem under consideration is now no longer self-adjoint. Nevertheless, using tools from analytic function theory, we will be able to obtain a partial answer to the question of when a knowledge of the transmission eigenvalues corresponding to spherically symmetric eigenfunctions uniquely determines the (spherically symmetric) index of refraction.