The higher-order fractional Schr\"odinger equation with nonlinear local perturbations: Uniqueness

TL;DR

This paper investigates the higher-order fractional Schr"odinger equation with nonlinear local perturbations, establishing well-posedness and demonstrating the unique determination of perturbations from boundary measurements, advancing inverse problem theory.

Contribution

It introduces new well-posedness results and proves the unique identifiability of nonlinear perturbations in higher-order fractional Schr"odinger equations.

Findings

Established Sobolev and H"older estimates for the nonlinear problem

Proved uniqueness of local nonlinear perturbations from Dirichlet-to-Neumann map

Applied higher-order linearization and unique continuation properties

Abstract

We study the higher-order fractional Schr\"odinger equation with local nonlinear perturbations and investigate both the forward and inverse problems. We establish both the Sobolev $H^s$ and H\"older $C^s$ estimates for the well-posedness of the nonlinear problem, based on the corresponding estimates derived for the linear fractional Schr\"odinger equation. For the inverse problem, we show that the local nonlinear perturbations can be uniquely determined from the Dirichlet-to-Neumann map, by using the higher-order linearization and the unique continuation property of the fractional Laplace operator.