Inverse spectral problems with sparse data and applications to passive imaging on manifolds

TL;DR

This paper develops a new spectral framework for passive imaging on manifolds, enabling the recovery of potentials and coefficients from sparse, partial spectral data without requiring orthogonality or norming constants.

Contribution

It introduces novel uniqueness results for inverse spectral problems with limited data on Riemannian manifolds, extending inverse spectral theory to highly incomplete, realistic measurements.

Findings

Unique recovery of potentials from partial spectral data.

Generic uniqueness for coefficients in evolutionary PDEs.

Framework applicable to passive imaging with sparse measurements.

Abstract

Motivated by inverse problems with a single passive measurement, we introduce and analyze a new class of inverse spectral problems on closed Riemannian manifolds. Specifically, we establish two general uniqueness results for the recovery of a potential in the stationary Schr\"odinger operator from partial spectral data, which consists of a possibly sparse subset of its eigenvalues and the restrictions of the corresponding eigenfunctions to a nonempty open subset of the manifold. Crucially, the eigenfunctions are not assumed to be orthogonal, and no information about global norming constants is required. The partial data formulation of our inverse spectral problems is naturally suited to the analysis of inverse problems with passive measurements, where only limited observational access to the solution is available. Leveraging this structure, we establish generic uniqueness results for a broad class of evolutionary PDEs, in which both the coefficients and the initial or source data are to be recovered from knowledge of the solution restricted to a subset of spacetime. These results introduce a spectral framework for passive imaging and extend inverse spectral theory into a regime characterized by highly incomplete, yet physically realistic, data.