Multidimensional Gel'fand Inverse Boundary Spectral Problem: Uniqueness and Stability

Abstract

The paper is devoted to the reconstruction of a compact Riemannian manifold from the Gel'fand boundary spectral data. These data consist of the eigenvalues and the boundary values of the eigenfunctions of the Laplace operator with the Neumann boundary condition. We provide the reconstruction procedure using the geometric variant of the boundary control method. In addition to the uniqueness and reconstruction results, we sketch recent developments in the conditional stability in this problem. These conditions are formulated in terms of some geometric restrictions traditional for the theory of geometric convergence.