The Gross-Pitaevskii hierarchy on general rectangular tori

TL;DR

This paper establishes a conditional uniqueness result for the Gross-Pitaevskii hierarchy on general rectangular tori, enabling a rigorous derivation of the nonlinear Schrödinger equation in certain cases, including irrational tori.

Contribution

It provides the first conditional uniqueness proof for the hierarchy on general tori, addressing an open problem for irrational tori and advancing the derivation of nonlinear Schrödinger equations.

Findings

Conditional uniqueness on rectangular tori

Derivation of nonlinear Schrödinger equation in 2D

Addresses open problem for irrational tori

Abstract

In this work, we study the Gross-Pitaevskii hierarchy on general --rational and irrational-- rectangular tori of dimension two and three. This is a system of infinitely many linear partial differential equations which arises in the rigorous derivation of the nonlinear Schr\"{o}dinger equation. We prove a conditional uniqueness result for the hierarchy. In two dimensions, this result allows us to obtain a rigorous derivation of the defocusing cubic nonlinear Schr\"{o}dinger equation from the dynamics of many-body quantum systems. On irrational tori, this question was posed as an open problem in previous work of Kirkpatrick, Schlein, and Staffilani.