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

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Consider a system of N bosons in three dimensions interacting via a repulsive short range pair potential N²V(N(xᵢ-xⱼ)), where =(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, Hₙᵏ ψₙ,₀ > ≤ Cᵏ Nᵏ for k=1,2,.... We also assume that the k-particle density matrices of the initial state are asymptotically factorized as N→∞. We prove that the k-particle density matrices of ψₙ,ₜ are also asymptotically factorized and the one particle orbital wave function solves the Gross-Pitaevskii equation, a cubic non-linear Schrödinger 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 ψₙ,₀ is assumed in a stronger sense.