Global uniqueness for the semilinear fractional Schr\"odinger equation

TL;DR

This paper proves that the unknown nonlinear term in a fractional semilinear Schr"odinger equation can be uniquely identified from boundary data across multiple dimensions, advancing inverse problem theory for nonlocal equations.

Contribution

It establishes global uniqueness for the inverse problem in fractional semilinear Schr"odinger equations for all dimensions greater than or equal to 2, including new comparison and estimate results.

Findings

Unique determination of $q(x,u)$ from Cauchy data.

Validity of comparison principle for the nonlocal equation.

Provision of an $L^ abla$ estimate under regularity assumptions.

Abstract

We study global uniqueness in an inverse problem for the fractional semilinear Schr\"{o}dinger equation $(-\Delta)^{s}u+q(x,u)=0$ with $s\in (0,1)$. We show that an unknown function $q(x,u)$ can be uniquely determined by the Cauchy data set. In particular, this result holds for any space dimension greater than or equal to $2$. Moreover, we demonstrate the comparison principle and provide a $L^\infty$ estimate for this nonlocal equation under appropriate regularity assumptions.