Derivation of the cubic NLS and Gross-Pitaevskii hierarchy from manybody dynamics in $d=3$ based on spacetime norms

TL;DR

This paper proves the convergence of the many-body Schrödinger dynamics to the cubic Gross-Pitaevskii hierarchy in three dimensions, using spacetime norm techniques, and derives the cubic NLS without assuming solution factorization.

Contribution

It provides the first rigorous derivation of the cubic GP hierarchy in 3D from many-body dynamics using spacetime norms, addressing a longstanding open problem.

Findings

Convergence of BBGKY hierarchy to GP hierarchy in 3D.

Derivation of cubic NLS from factorized solutions.

Application of spacetime norm methods to hierarchy analysis.

Abstract

We derive the defocusing cubic Gross-Pitaevskii (GP) hierarchy in dimension $d=3$, from an $N$-body Schr\"{o}dinger equation describing a gas of interacting bosons in the GP scaling, in the limit $N\rightarrow\infty$. The main result of this paper is the proof of convergence of the corresponding BBGKY hierarchy to a GP hierarchy in the spaces introduced in our previous work on the well-posedness of the Cauchy problem for GP hierarchies, \cite{chpa2,chpa3,chpa4}, which are inspired by the solutions spaces based on space-time norms introduced by Klainerman and Machedon in \cite{klma}. We note that in $d=3$, this has been a well-known open problem in the field. While our results do not assume factorization of the solutions, consideration of factorized solutions yields a new derivation of the cubic, defocusing nonlinear Schr\"odinger equation (NLS) in $d=3$.