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 $N^{2}V(N(x_i-x_j))$, where $\bx=(x_1, >…, x_N)$denotesthepositionsoftheparticles\mathrm{.}Let$H_N$ denote the Hamiltonian of the system and let ${\psi }_{N,t}$ be the solution to the Schr"odinger equation. Suppose that the initial data ${\psi }_{N,0}$ satisfies the energy condition $$<{\psi }_{N,0},H_N{\psi }_{N,0}>\le CN>\mathrm{.}$$ and that the one-particle density matrix converges to a projection as $N\to \mathrm{\infty }$. Then, we prove that the $k$-particle density matrices of ${\psi }_{N,t}$ factorize in the limit $N\to \mathrm{\infty }$. Moreover, the one particle orbital wave function solves the time-dependent Gross-Pitaevskii equation, a cubic non-linear Schr"odinger 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.