Gross-Pitaevskii Equation as the Mean Field Limit of Weakly Coupled Bosons

TL;DR

This paper proves that the dynamics of large boson systems with weak interactions converge to the Gross-Pitaevskii hierarchy, establishing a link between many-body quantum dynamics and nonlinear Schrödinger equations under specific scaling limits.

Contribution

It demonstrates the convergence of $k$-particle density matrices of boson systems to solutions of the GP hierarchy for a specific interaction scaling, advancing understanding of mean field limits.

Findings

Limit points of density matrices satisfy the GP hierarchy.

Convergence holds for interaction parameter $a = N^{- ext{eps}}$ with $0< ext{eps}<3/5$.

The coupling constant in the GP equation is given by the integral of the potential.

Abstract

We consider the dynamics of $N$ boson systems interacting through a pair potential $N^{-1} V_a(x_i-x_j)$ where $V_a (x) = a^{-3} V (x/a)$. We denote the solution to the $N$-particle Schr\"odinger equation by $\psi_{N, t}$. Recall that the Gross-Pitaevskii (GP) equation is a nonlinear Schr\"odinger equation and the GP hierarchy is an infinite BBGKY hierarchy of equations so that if $u_t$ solves the GP equation, then the family of $k$-particle density matrices $\{\otimes_k u_t, k\ge 1 \}$ solves the GP hierarchy. Under the assumption that $a = N^{-\eps}$ for $0 < \eps < 3/5$, we prove that as $N\to \infty$ the limit points of the $k$-particle density matrices of $\psi_{N,t}$ are solutions of the GP hierarchy with the coupling constant in the nonlinear term of the GP equation given by $\int V(x) dx$. The uniqueness of the solutions to this hierarchy remains an open question.