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

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 $λ_j$ of the Schrödinger operator $-Δ_g + q$ and the restriction of the eigenfunctions $ϕ_j|_U$ on a given open subset $U\subset M$, where $Δ_g$ 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.