On the Calder\'on problem for nonlocal Schr\"odinger equations with homogeneous, directionally antilocal principal symbols

TL;DR

This paper investigates the Calderón problem for nonlocal Schr"odinger equations with directionally supported kernels, establishing uniqueness results for inverse problems under new geometric conditions due to directional antilocality.

Contribution

It introduces a framework for inverse problems involving nonlocal operators with directional support, providing new uniqueness results and geometric conditions for measurements.

Findings

Unique determination of potentials from single measurements under directional antilocality.

New geometric conditions on measurement domains for inverse problems.

Differences in properties between symmetric and non-symmetric nonlocal operators.

Abstract

In this article we consider direct and inverse problems for $\alpha$-stable, elliptic nonlocal operators whose kernels are possibly only supported on cones and which satisfy the structural condition of \emph{directional antilocality} as introduced in \cite{I86}. We consider the Dirichlet problem for these operators on the "domain of dependence of the operator" and in several, adapted function spaces. This formulation allows one to avoid natural "gauges" which would else have to be considered in the study of the associated inverse problems. Exploiting the directional antilocality of these operators we complement the investigation of the \emph{direct problem} with infinite data and single measurement uniqueness results for the associated \emph{inverse problems}. Here, due to the only directional antilocality, new geometric conditions arise on the measurement domains. We discuss both the setting of symmetric and a particular class of non-symmetric nonlocal elliptic operators, and contrast the corresponding results for the direct and inverse problems. In particular for only "one-sided operators" new phenomena emerge both in the direct and inverse problems: For instance, it is possible to study the problem in data spaces involving local and nonlocal data, the unique continuation property may not hold in general and further restrictions on the measurement set for the inverse problem arise.