Determining Schrodinger Potentials on Riemannian Manifolds

Abstract

We consider the inverse problem of determining a potential for the stationary Schr\"odinger equation on certain Riemannian manifolds with boundary from the Dirichlet-to-Neumann map. We derive two unique identifiability results. The first assumes that the Riemannian metric is close to the Euclidean metric, with the closeness quantified by a relative Fourier condition imposed on the potential function. The second assumes that the manifold is of product type and that the potential function is additively separable. Both the results and the technical arguments are new, contributing to the advancement of this challenging open inverse problem.