László Erdös, Benjamin Schlein, Horng-Tzer Yau
Derivation of the Gross-Pitaevskii Equation for the Dynamics of Bose-Einstein Condensate
Preprint series: ESI preprints

MSC:

81V70 Many-body theory

81T18 Feynman diagrams

35Q55 NLS-like (nonlinear Schrodinger) equations, See Also {

Abstract: Consider a system of $N$ bosons in three dimensions interacting via
a repulsive short range pair potential $N^2V(N(x_i-x_j))$, where
$\bx=(x_1, \ldots, x_N)$ denotes the positions of the particles. 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
\[ \langle
\psi_{N,0}, H_N^k \psi_{N,0} \rangle \leq C^k N^k \;
\]
for $k=1,2,\ldots $.
We also assume that
the $k$-particle density matrices of the initial state are
asymptotically
factorized as $N\to\infty$.
We prove that
the $k$-particle density matrices of $\psi_{N,t}$ are also
asymptotically
factorized and the one particle orbital wave function solves the
Gross-Pitaevskii equation, a cubic non-linear Schr\"odinger equation
with the coupling constant given by the scattering length of the
potential $V$. We also prove the same conclusion if the energy
condition holds only for $k=1$ but the factorization of $\psi_{N,0}$
is assumed in a stronger sense.


Keywords: Nonlinear Schrodinger equation, interacting bosons, Bose-Einstein condensate