Uniqueness and reconstruction for the fractional Calder\'on problem with a single measurement

TL;DR

This paper proves that it is possible to uniquely determine an unknown potential in the fractional Calderón problem using only a single measurement, extending previous results that required infinitely many measurements, and provides a constructive method for reconstruction.

Contribution

It establishes global uniqueness with a single measurement in the fractional Calderón problem and introduces a constructive reconstruction procedure based on unique continuation principles.

Findings

Unique determination of potential from one measurement

Constructive reconstruction method provided

Applicable to arbitrary, possibly disjoint exterior data sets

Abstract

We show global uniqueness in the fractional Calder\'on problem with a single measurement and with data on arbitrary, possibly disjoint subsets of the exterior. The previous work \cite{GhoshSaloUhlmann} considered the case of infinitely many measurements. The method is again based on the strong uniqueness properties for the fractional equation, this time combined with a unique continuation principle from sets of measure zero. We also give a constructive procedure for determining an unknown potential from a single exterior measurement, based on constructive versions of the unique continuation result that involve different regularization schemes.