Inverse Spectral Problem With Low Regularity Refractive Index

TL;DR

This paper studies the inverse spectral problem for radial refractive indices, showing non-uniqueness for certain regularity classes and establishing conditions for unique determination using spectral data.

Contribution

It demonstrates non-uniqueness for piecewise twice differentiable indices and proves uniqueness for smoother indices with additional a priori information.

Findings

Non-uniqueness for piecewise twice differentiable n

Uniqueness for twice continuously differentiable n with partial info

Spectral data can determine n under certain regularity conditions

Abstract

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.