The Calder\'on problem for the fractional Schr\"odinger equation with drift

TL;DR

This paper proves the unique simultaneous determination of drift and potential in a fractional Schr"odinger equation from exterior measurements, establishing stability and finite measurement reconstruction methods, with implications for hybrid inverse problems.

Contribution

It establishes the first uniqueness and stability results for the fractional Schr"odinger Calderón problem with drift, including finite measurement reconstruction algorithms.

Findings

Unique determination of drift and potential from exterior data

Logarithmic stability estimate under certain assumptions

Finite measurement reconstruction algorithm

Abstract

We investigate the Calder\'on problem for the fractional Schr\"odinger equation with drift, proving that the unknown drift and potential in a bounded domain can be determined simultaneously and uniquely by an infinite number of exterior measurements. In particular, in contrast to its local analogue, this nonlocal problem does \emph{not} enjoy a gauge invariance. The uniqueness result is complemented by an associated logarithmic stability estimate under suitable apriori assumptions. Also uniqueness under finitely many \emph{generic} measurements is discussed. Here the genericity is obtained through \emph{singularity theory} which might also be interesting in the context of hybrid inverse problems. Combined with the results from \cite{GRSU18}, this yields a finite measurements constructive reconstruction algorithm for the fractional Calder\'on problem with drift. The inverse problem is formulated as a partial data type nonlocal problem and it is considered in any dimension $n\geq 1$.