On the Uniqueness of a Spherically Symmetric Speed of Sound from Transmission Eigenvalues

Consider the inverse acoustic scattering problem for spherically symmetric inhomogeneity of compact support. Define the corresponding homogeneous and inhomogeneous interior transmission problems. We study the subset of transmission eigenvalues corresponding to spherically symmetric eigenfunctions of the homogeneous interior transmission problem. These transmission eigenvalues are shown to be zeros of the scattering amplitude and also the set of eigenvalues of a special Sturm-Liouville problem. A uniqueness theorem for the potential of the derived Sturm Liouville problem is proved when the data are the given spectra and partial knowledge of the potential. A corollary of this theorem is a uniqueness theorem for the original inverse acoustic scattering problem.