Multidimensional Borg–Levinson theorem

Abstract

We consider the inverse problem of the reconstruction of a Schrödinger operator on an unknown Riemannian manifold or a domain of Euclidean space. The data used are a part of the boundary Σ and the eigenvalues corresponding to a set of impedances in the Robin boundary condition which vary on Σ. The proof is based on the analysis of the behaviour of the eigenfunctions on the boundary as well as the perturbation theory of eigenvalues. These reduces the problem to an inverse boundary spectral problem solved by the boundary control method.