  We consider the dynamics of $N$ boson systems interacting through a pair
potential $N^{-1} V_a(x_i-x_j)$ where $V_a (x) = a^{-3} V (x/a)$. We denote the
solution to the $N$-particle Schr\"odinger equation by $\psi_{N, t}$. Recall
that the Gross-Pitaevskii (GP) equation is a nonlinear Schr\"odinger equation
and the GP hierarchy is an infinite BBGKY hierarchy of equations so that if
$u_t$ solves the GP equation, then the family of $k$-particle density matrices
$\{\otimes_k u_t, k\ge 1 \}$ solves the GP hierarchy. Under the assumption that
$a = N^{-\eps}$ for $0 < \eps < 3/5$, we prove that as $N\to \infty$ the limit
points of the $k$-particle density matrices of $\psi_{N,t}$ are solutions of
the GP hierarchy with the coupling constant in the nonlinear term of the GP
equation given by $\int V(x) dx$. The uniqueness of the solutions to this
hierarchy remains an open question.
