Calder\'{o}n problem for fractional Schr\"{o}dinger operators on closed Riemannian manifolds

TL;DR

This paper addresses the inverse problem of determining a Riemannian manifold and potential from fractional Schr"odinger operator data, introducing new methods that connect spectral theory, nonlocal equations, and inverse problems.

Contribution

It introduces a novel approach to the Calderón problem for fractional Schr"odinger operators on closed manifolds, including a new spectral problem variant and an entanglement principle.

Findings

Unique determination of manifold and potential from Cauchy data

Development of a new spectral inverse problem without energy normalization

Discovery of an entanglement principle for nonlocal equations

Abstract

We study an analog of the anisotropic Calder\'on problem for fractional Schr\"odinger operators $(-\Delta_g)^\alpha + V$ with $\alpha \in (0,1)$ on closed Riemannian manifolds of dimensions two and higher. We prove that the knowledge of a Cauchy data set of solutions of the fractional Schr\"odinger equation, given on an open nonempty a priori known subset of the manifold determines both the Riemannian manifold up to an isometry and the potential up to the corresponding gauge transformation, under certain geometric assumptions on the manifold as well as the observation set. Our method of proof is based on: (i) studying a new variant of the Gel'fand inverse spectral problem without the normalization assumption on the energy of eigenfunctions, and (ii) the discovery of an entanglement principle for nonlocal equations involving two or more compactly supported functions. Our solution to (i) makes connections to antipodal sets as well as local control for eigenfunctions and quantum chaos, while (ii) requires sharp interpolation results for holomorphic functions. We believe that both of these results can find applications in other areas of inverse problems.