On Single Measurement Stability for the Fractional Calder\'on Problem

TL;DR

This paper establishes the logarithmic stability of the single measurement inverse problem for the fractional Calderón problem, enhancing understanding of solution sensitivity and uniqueness in fractional inverse problems.

Contribution

It proves the logarithmic stability of the single measurement inverse problem for the fractional Calderón problem, building on previous uniqueness results and boundary doubling estimates.

Findings

Logarithmic stability of the fractional Calderón problem with a single measurement.

Control of the order of vanishing of solutions to the fractional Schrödinger equation.

Extension of stability results using boundary doubling estimates.

Abstract

In this short note we prove the logarithmic stability of the single measurement uniqueness result for the fractional Calder\'on problem which had been derived in \cite{GRSU18}. To this end, we use the quantitative uniqueness results established in \cite{RS20a} and complement these bounds with a boundary doubling estimate. The latter yields control of the order of vanishing of solutions to the fractional Schr\"odinger equation. Then, following a scheme introduced in \cite{S10,ASV13} in the context of the determination of a surface impedance from far field measurements, this allows us to deduce logarithmic stability of the potential $q$.