The Rigorous Derivation of the 2D Cubic Focusing NLS from Quantum Many-body Evolution

TL;DR

This paper rigorously derives the 2D focusing nonlinear Schrödinger equation as the mean-field limit of a quantum many-body system of bosons with attractive interactions, establishing new stability estimates and convergence results.

Contribution

It introduces a novel stability estimate for attractive interactions and proves the convergence of the quantum dynamics to the focusing NLS in 2D.

Findings

Established energy control by the Sobolev norm for attractive interactions.

Proved convergence of the BBGKY hierarchy to the focusing NLS.

Derived stability estimates using a quantum de Finetti theorem.

Abstract

We consider a 2D time-dependent quantum system of $N$-bosons with harmonic external confining and \emph{attractive} interparticle interaction in the Gross-Pitaevskii scaling. We derive stability of matter type estimates showing that the $k$-th power of the energy controls the $H^{1}$ Sobolev norm of the solution over $k$-particles. This estimate is new and more difficult for attractive interactions than repulsive interactions. For the proof, we use a version of the finite-dimensional quantum di Finetti theorem from [49]. A high particle-number averaging effect is at play in the proof, which is not needed for the corresponding estimate in the repulsive case. This a priori bound allows us to prove that the corresponding BBGKY hierarchy converges to the GP limit as was done in many previous works treating the case of repulsive interactions. As a result, we obtain that the \emph{focusing} nonlinear Schr\"{o}dinger equation is the mean-field limit of the 2D time-dependent quantum many-body system with attractive interatomic interaction and asymptotically factorized initial data. An assumption on the size of the $L^{1}$-norm of the interatomic interaction potential is needed that corresponds to the sharp constant in the 2D Gagliardo-Nirenberg inequality though the inequality is not directly relevant because we are dealing with a trace instead of a power.