Reconstruction and stability in Gel’fand’s inverse interior spectral problem

Assume that M is a compact Riemannian manifold of bounded geometry given by restrictions on its diameter, Ricci curvature and injectivity radius. Assume we are given, with some error, the first eigenvalues of the Laplacian ∆ on M as well as the corresponding eigenfunctions restricted on an open set in M . We then construct a stable approximation to the manifold (M,g). Namely, we construct a metric space and a Riemannian manifold which differ, in a proper sense, just a little from M when the above data are given with a small error. We give an explicit logarithmic stability estimate on how the constructed manifold and the metric on it depend on the errors in the given data. Moreover a similar stability estimate is derived for the Gel’fand’s inverse problem. The proof is based on methods from geometric convergence, a quantitative stability estimate for the unique continuation and a new version of the geometric Boundary Control method.