Derivation of the Gross-Pitaevskii Equation for the Dynamics of Bose-Einstein Condensate

TL;DR

This paper rigorously derives the Gross-Pitaevskii equation as the effective dynamics for Bose-Einstein condensates in the mean-field limit, connecting many-body quantum mechanics to a nonlinear Schrödinger equation.

Contribution

It provides a rigorous proof that the many-body Schrödinger dynamics converges to the Gross-Pitaevskii equation under certain initial conditions and energy bounds.

Findings

The k-particle density matrices become asymptotically factorized over time.

The one-particle wave function satisfies the cubic nonlinear Schrödinger equation.

The coupling constant is given by the scattering length of the interaction potential.

Abstract

Consider a system of $N$ bosons in three dimensions interacting via a repulsive short range pair potential $N^2V(N(x_i-x_j))$, where $\bx=(x_1, >..., x_N)$ denotes the positions of the particles. Let $H_N$ denote the Hamiltonian of the system and let $\psi_{N,t}$ be the solution to the Schr\"odinger equation. Suppose that the initial data $\psi_{N,0}$ satisfies the energy condition \[ < \psi_{N,0}, H_N^k \psi_{N,0} > \leq C^k N^k \] for $k=1,2,... $. We also assume that the $k$-particle density matrices of the initial state are asymptotically factorized as $N\to\infty$. We prove that the $k$-particle density matrices of $\psi_{N,t}$ are also asymptotically factorized and the one particle orbital wave function solves the Gross-Pitaevskii equation, a cubic non-linear Schr\"odinger equation with the coupling constant given by the scattering length of the potential $V$. We also prove the same conclusion if the energy condition holds only for $k=1$ but the factorization of $\psi_{N,0}$ is assumed in a stronger sense.