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

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 $ψ_{N,t}$ be the solution to the Schrödinger equation. Suppose that the initial data $ψ_{N,0}$ satisfies the energy condition $< ψ_{N,0}, H_N ψ_{N,0} > \leq C N >.$ 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 $ψ_{N,t}$ 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 Schrödinger 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.

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