Inverse spectral problem with low regularity refractive index

This article investigates the unique determination of a radial refractive index $ n $ from spectral data. The first part of our analysis shows that piecewise constant refractive indices cannot be uniquely recovered from the special transmission eigenvalues associated with radially symmetric eigenfunctions. In the second part, we obtain uniqueness results for indices in the classes $ C_p^2[0, 1]\cap\mathcal{M} $ and $ C^{1, 1}[0, 1]\cap\mathcal{M} $, provided that the complete set of special transmission eigenvalues is augmented by partial a priori information. Consequently, our findings extend prior uniqueness results to multilayered media with arbitrarily many jumps and remove the global assumptions $ n < 1 $ or $ n > 1 $, though they rely on the supplementary conditions specified in this work.