Two theorems show that a potential on a closed Riemannian manifold is uniquely recovered from sparse eigenvalue data and eigenfunction restrictions to an open set, with applications to single-measurement passive imaging.
Stability of Gel'fand's inverse interior spectral problem for Schr\"odinger operators
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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.
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Inverse spectral problems with sparse data and applications to passive imaging on manifolds
Two theorems show that a potential on a closed Riemannian manifold is uniquely recovered from sparse eigenvalue data and eigenfunction restrictions to an open set, with applications to single-measurement passive imaging.