An inverse problem for the fractional Schr\"odinger equation in a magnetic field

TL;DR

This paper proves that the electromagnetic field in a fractional magnetic Schr"odinger equation can be uniquely identified from exterior measurements, extending previous results and connecting to long jump random walks.

Contribution

It establishes global uniqueness in an inverse problem for FMSE using Alessandrini's identity and Runge approximation, generalizing prior fractional Laplacian work.

Findings

Unique determination of electromagnetic fields from exterior data

Extension of inverse problem techniques to fractional magnetic Schr"odinger equations

Connection between FMSE and long jump random walk models

Abstract

This paper shows global uniqueness in an inverse problem for a fractional magnetic Schr\"odinger equation (FMSE): an unknown electromagnetic field in a bounded domain is uniquely determined up to a natural gauge by infinitely many measurements of solutions taken in arbitrary open subsets of the exterior. The proof is based on Alessandrini's identity and the Runge approximation property, thus generalizing some previous works on the fractional Laplacian. Moreover, we show with a simple model that the FMSE relates to a long jump random walk with weights.