Uniqueness for the inverse spectral modifiedtransmission eigenvalue problem for a piecewisecontinuous refractive index

In this work, we investigate the uniqueness of the inverse spectral modified transmission eigenvalue problem in the special case where the refractive index is piecewise continuous and spherically symmetric. The direct modified transmission eigenvalue problem arises as a different eigenvalue problem involving a spectral parameter only in the Helmholtz equation corresponding to an artificial metamaterial background. We prove that one eigenvalue from each characteristic function, derived from separation of variables, is sufficient to provide a ratio of the one-dimensional Cauchy data of the eigenfunctions. This is equivalent to the knowledge of the Dirichlet-to-Neumann map for the Helmholtz equation in the interior domain with the same refractive index. We then show that the refractive index can be uniquely determined by employing results from the Borg-Levinson theory for Shr"ondinger equation. We also suggest a technique for classifying eigenvalues according to their corresponding characteristic function without any knowledge of the refractive index. Moreover, we prove a monotonicity result for specific spectral functions derived from the characteristics, which validates the uniqueness for piecewise continuous refractive indices. Finally, we present some analytical examples of piecewise constant refractive indices with one jump and their recovering relationship with a set of few modified transmission eigenvalues.