Inverse Spectral Problem With Low Regularity Refractive Index

This article investigates the unique determination of a radial refractive index n from spectral data. First, we demonstrate that for piecewise twice continuously differentiable functions, n is not uniquely determined by the special transmission eigenvalues associated with radially symmetric eigenfunctions. Subsequently we prove that if n \in M is twice continuously differentiable functions(or continuously differentiable functions with Lipschitz continuous derivative), then n is uniquely determined on [0,1] by all special transmission eigenvalues when supplemented by partial a priori information on the refractive index.