Stability of Gel'fand's inverse interior spectral problem for Schrödinger operators

Abstract

We study Gel'fand's inverse interior spectral problem of determining a closed Riemannian manifold $(M,g)$ and a potential function $q$ from the knowledge of the eigenvalues ${\lambda }_j$ of the Schrödinger operator $-{\Delta }_g+q$ and the restriction of the eigenfunctions ${\varphi }_j{|}_U$ on a given open subset $U\subset M$, where ${\Delta }_g$ is the Laplace-Beltrami operator on $(M,g)$. We prove that an approximation of finitely many spectral data on $U$ determines a finite metric space that is close to $(M,g)$ in the Gromov-Hausdorff topology, and further determines a discrete function that approximates the potential $q$ with uniform estimates. This leads to a quantitative stability estimate for the inverse interior spectral problem for Schrödinger operators in the general case.

29 pages, 2 figures