 04333 Alexander Elgart, Laszlo Erdos, Benjamin Schlein, HorngTzer Yau
 GrossPitaevskii Equation as the Mean Field Limit of Weakly
Coupled Bosons
(60K, LateX )
Oct 25, 04

Abstract ,
Paper (src),
View paper
(auto. generated ps),
Index
of related papers

Abstract. We consider the dynamics of $N$ boson systems interacting
through a pair potential $N^{1} V_a(x_ix_j)$ 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 GrossPitaevskii (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.
 Files:
04333.src(
04333.comments ,
04333.keywords ,
gpe.tex )