The higher order fractional Calder\'on problem for linear local operators: uniqueness

TL;DR

This paper proves the unique recovery of linear local operator coefficients in a fractional Schrödinger equation from boundary measurements, extending previous results to higher order perturbations.

Contribution

It introduces a method to uniquely determine higher order linear PDO coefficients in fractional Schrödinger equations from Dirichlet-to-Neumann data.

Findings

Unique recovery of PDO coefficients from DN map.

Extension of results to higher order perturbations.

Applicable to coefficients in Sobolev multiplier and fractional Sobolev spaces.

Abstract

We study an inverse problem for the fractional Schr\"odinger equation (FSE) with a local perturbation by a linear partial differential operator (PDO) of order smaller than the order of the fractional Laplacian. We show that one can uniquely recover the coefficients of the PDO from the Dirichlet-to-Neumann (DN) map associated to the perturbed FSE. This is proved for two classes of coefficients: coefficients which belong to certain spaces of Sobolev multipliers and coefficients which belong to fractional Sobolev spaces with bounded derivatives. Our study generalizes recent results for the zeroth and first order perturbations to higher order perturbations.