Unconditional Uniqueness Results for the Nonlinear Schr\"odinger Equation

TL;DR

This paper establishes new unconditional uniqueness results for the cubic nonlinear Schr"odinger equation on bounded domains, especially rectangular tori, by linking solutions to the Gross-Pitaevskii hierarchy and analyzing well-posedness in Fourier-Lebesgue spaces.

Contribution

Introduces a novel approach to prove unconditional uniqueness for cubic NLS on bounded domains using hierarchy analysis and Fourier-Lebesgue space well-posedness.

Findings

Unconditional uniqueness results cover the full scaling-subcritical regime in high dimensions.

New strategy applicable to solutions on rectangular tori.

Well-posedness in Fourier-Lebesgue spaces implies unconditional uniqueness.

Abstract

We study the problem of unconditional uniqueness of solutions to the cubic nonlinear Schr\"odinger equation. We introduce a new strategy to approach this problem on bounded domains, in particular on rectangular tori. It is a known fact that solutions to the cubic NLS give rise to solutions of the Gross-Pitaevskii hierarchy, which is an infinite-dimensional system of linear equations. By using the uniqueness analysis of the Gross-Pitaevskii hierarchy, we obtain new unconditional uniqueness results for the cubic NLS on rectangular tori, which cover the full scaling-subcritical regime in high dimensions. In fact, we prove a more general result which is conditional on the domain. In addition, we observe that well-posedness of the cubic NLS in Fourier-Lebesgue spaces implies unconditional uniqueness.