Rigorous Derivation of the Gross-Pitaevskii Equation with a Large Interaction Potential

TL;DR

This paper rigorously derives the Gross-Pitaevskii equation from many-body quantum mechanics for bosons with large, short-range interactions, removing previous restrictions on the potential size.

Contribution

The authors develop a new method to derive the Gross-Pitaevskii equation without assuming small interaction potentials, extending previous results.

Findings

Proved convergence of k-particle density matrices to factorized form.

Showed the one-particle wave function satisfies the Gross-Pitaevskii equation.

Removed size restrictions on 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 \[ < \psi_{N,0}, H_N \psi_{N,0} > \leq C N >. \] and that the one-particle density matrix converges to a projection as $N \to \infty$. Then, we prove that the $k$-particle density matrices of $\psi_{N,t}$ factorize in the limit $N \to \infty$. Moreover, the one particle orbital wave function solves the time-dependent Gross-Pitaevskii equation, a cubic non-linear Schr\"odinger equation with the coupling constant proportional to the scattering length of the potential $V$. In \cite{ESY}, we proved the same statement under the condition that the interaction potential $V$ is sufficiently small; in the present work we develop a new approach that requires no restriction on the size of the potential.