Uniqueness for the fractional Calder\'on problem with quasilocal perturbations

TL;DR

This paper investigates the fractional Schr"odinger equation with quasilocal perturbations, establishing unique continuation, Runge approximation, and inverse problem solutions under decay conditions, extending recent fractional Calder"on problem results.

Contribution

It introduces new uniqueness results for the inverse fractional Calder"on problem with quasilocal perturbations, generalizing prior locally perturbed cases.

Findings

Unique continuation and Runge approximation hold with decay assumptions.

Quantitative stability results via propagation of smallness.

Uniqueness in recovering quasilocal perturbations from DN data.

Abstract

We study the fractional Schr\"odinger equation with quasilocal perturbations. These are a family of nonlocal perturbations vanishing at infinity, which include e.g. convolutions against Schwartz functions. We show that the qualitative unique continuation and Runge approximation properties hold in the assumption of sufficient decay. Quantitative versions of both results are also obtained via a propagation of smallness analysis for the Caffarelli-Silvestre extension. The results are then used to show uniqueness in the inverse problem of retrieving a quasilocal perturbation from DN data under suitable geometric assumptions. Our work generalizes recent results regarding the locally perturbed fractional Calder\'on problem.