An inverse problem for the fractional Schrödinger equation in a magnetic field

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.