Monotonicity-based inversion of the fractional Schr\"odinger equation I. Positive potentials

TL;DR

This paper introduces a monotonicity-based approach to uniquely identify positive potentials and obstacles in the fractional Schrödinger equation, providing constructive proofs and a dimension-independent reconstruction method.

Contribution

It establishes if-and-only-if monotonicity relations for positive potentials and their Dirichlet-to-Neumann maps, enabling constructive uniqueness proofs and obstacle reconstruction.

Findings

Proves monotonicity relations between potentials and Dirichlet-to-Neumann maps.

Provides a constructive proof of uniqueness for the nonlocal Calderón problem.

Develops a dimension-independent obstacle reconstruction method.

Abstract

We consider the inverse problems of for the fractional Schr\"odinger equation by using monotonicity formulas. We provide if-and-only-if monotonicity relations between positive bounded potentials and their associated nonlocal Dirichlet-to-Neumann maps. Based on the monotonicity relation, we can prove uniqueness for the nonlocal Calder\'on problem in a constructive manner. Secondly, we offer a reconstruction method for an unknown obstacles in a given domain. Our method is independent of the dimension $n\geq 2$ and only requires the background solution of the fractional Schr\"odinger equation.