 05365 Laszlo Erdos, Benjamin Schlein, HorngTzer Yau
 Derivation of the GrossPitaevskii Hierarchy for the
Dynamics of BoseEinstein Condensate
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Oct 21, 05

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Abstract. Consider a system of $N$ bosons on the three dimensional unit
torus interacting via a pair potential $N^2V(N(x_ix_j))$, 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 GrossPitaevskii (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 interparticle
distance. Our proof can be extended to a modified Hamiltonian
which only forbids at least $n$ particles from coming close
together, for any fixed $n$.
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