The inverse transmission eigenvalue problem for a discontinuous refractive index

We consider the inverse spectral problem of determining a spherically symmetric and discontinuous refractive index n(r) from interior transmission eigenvalues. Using Liouville's transform, we investigate the asymptotic properties of the solution of an auxiliary initial value problem for large wave numbers and the asymptotic behaviour of the characteristic determinants derived from the eigenfunction expansions. Next, we assume that we know all transmission eigenvalues with spherically symmetric eigenfunctions and prove under some conditions that the transformed discontinuity of the refractive index can be determined. Finally we prove that the knowledge of all transmission eigenvalues including multiplicities uniquely determines n(r), under the assumption that n(0) is known and either n(r) > 1 or 0 < n(r) < 1 by using a moment type result and applying Müntz's theorem.