Stability of Gel'fand's inverse interior spectral problem for Schr\"odinger operators

TL;DR

This paper establishes a quantitative stability estimate for the inverse interior spectral problem of Schr"odinger operators on Riemannian manifolds, showing that spectral data approximations can recover the manifold and potential with controlled accuracy.

Contribution

It provides the first stability result quantifying how spectral data approximations determine the manifold and potential in the inverse problem.

Findings

Finite spectral data approximations determine a close metric space in Gromov-Hausdorff topology.

Spectral data approximations allow for a discrete potential function with uniform estimates.

The results yield a quantitative stability estimate for the inverse problem.

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 Schr\"odinger operator $-\Delta_g + q$ and the restriction of the eigenfunctions $\phi_j|_U$ 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 Schr\"odinger operators in the general case.