The Gross–Pitaevskii Hierarchy on General Rectangular Tori

Abstract

In this work, we study the Gross–Pitaevskii hierarchy on general—rational and irrational—rectangular tori of dimensions two and three. This is a system of infinitely many linear partial differential equations which arises in the rigorous derivation of the nonlinear Schrödinger 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 Schrödinger equation from the dynamics of many-body quantum systems. On irrational tori, this question was posed as an open problem in the previous work of Kirkpatrick, Schlein, and Staffilani.