Laszlo Erdos, Benjamin Schlein, Horng-Tzer Yau
Derivation of the Gross-Pitaevskii Hierarchy for the 
Dynamics of Bose-Einstein Condensate
(209K, Latex)

ABSTRACT.  Consider a system of $N$ bosons on the three dimensional unit 
torus interacting via a pair potential $N^2V(N(x_i-x_j))$, where 
$\bx=(x_1, \ldots, x_N)$ denotes the positions of the particles. 
Suppose that the initial data $\psi_{N,0}$ satisfies the condition 
\[ 
\langle \psi_{N,0}, H_N^2 \psi_{N,0} \rangle \leq C N^2 
\] 
where $H_N$ is the Hamiltonian of the Bose system. This condition 
is satisfied if $\psi_{N,0}= W_N \phi_{N,0}$ where $W_N$ is an 
approximate ground state to $H_N$ and $\phi_{N,0}$ is regular. Let 
$\psi_{N,t}$ denote the solution to the Schr\"odinger equation 
with Hamiltonian $H_N$. Gross and Pitaevskii proposed to model 
the dynamics of such system by a nonlinear Schr\"odinger equation, 
the Gross-Pitaevskii (GP) equation. 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. 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. The 
uniqueness of the solutions to this hierarchy remains an open 
question. Our analysis requires that the $N$ boson dynamics is 
described by a modified Hamiltonian which cuts off the pair 
interactions whenever at least three particles come into a region 
with diameter much smaller than the typical inter-particle 
distance. Our proof can be extended to a modified Hamiltonian 
which only forbids at least $n$ particles from coming close 
together, for any fixed $n$.