Rigorous derivation of the Gross-Pitaevskii equation with a large interaction potential

Abstract

Consider a system of N N bosons in three dimensions interacting via a repulsive short range pair potential N 2 V ( N ( x i − x j ) ) N^2V(N(x_i-x_j)) , where x = ( x 1 , … , x N ) \mathbf {x}=(x_1, \ldots , x_N) denotes the positions of the particles. Let H N H_N denote the Hamiltonian of the system and let ψ N , t \psi _{N,t} be the solution to the Schrödinger equation. Suppose that the initial data ψ N , 0 \psi _{N,0} satisfies the energy condition \[ ⟨ ψ N , 0 , H N ψ N , 0 ⟩ ≤ C N \langle \psi _{N,0}, H_N \psi _{N,0} \rangle \leq C N \] and that the one-particle density matrix converges to a projection as N → ∞ N \to \infty . Then, we prove that the k k -particle density matrices of ψ N , t \psi _{N,t} factorize in the limit N → ∞ 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 V . In a recent paper, we proved the same statement under the condition that the interaction potential V V is sufficiently small. In the present work we develop a new approach that requires no restriction on the size of the potential.