Rigorous Derivation of the Gross-Pitaevskii Equation: the Case of a Strong Potential

Consider a system of N bosons in three dimensions interacting via a repulsive short range pair potential NV (N(xi − xj)), where x = (x1, . . . , xN ) denotes the positions of the particles. Let HN denote the Hamiltonian of the system and let ψN,t be the solution to the Schrödinger equation. Suppose that the initial data ψN,0 satisfies the energy condition 〈ψN,0, HNψN,0〉 ≤ CN . and that the one-particle density matrix converges to a projection as N → ∞. Then, we prove that the k-particle density matrices of ψN,t factorize in the limit N → ∞. 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 [10], we proved the same statement under the condition that the interaction potential V is sufficiently small; in the present work we extend this result to large potentials.