Rigorous derivation of the Gross-Pitaevskii equation with a large 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 $\mathbf {x}=(x_1, \ldots , x_N)$ 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 Schrödinger equation. Suppose that the initial data $\psi _{N,0}$ satisfies the energy condition \[ \langle \psi _{N,0}, H_N \psi _{N,0} \rangle \leq C N \] 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 nonlinear Schrödinger equation with the coupling constant proportional to the scattering length of the potential $V$. In a recent paper, 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.