On the Time Dependent Gross Pitaevskii- and Hartree Equation

Abstract

We are interested in solutions ${\Psi }_t$ of the Schrödinger equation of $N$ interacting bosons under the influence of a time dependent external field, where the range and the coupling constant of the interaction scale with $N$ in such a way, that the interaction energy per particle stays more or less constant. Let ${\mathcal{N}}^{{\varphi }_0}$ be the particle number operator with respect to some ${\varphi }_0\in L^2({\mathbb{R}}^3\to \mathbb{C})$. Assume that the relative particle number of the initial wave function $N^{-1}<{\Psi }_0,{\mathcal{N}}^{{\varphi }_0}{\Psi }_0>$ converges to one as $N\to \infty$. We shall show that we can find a ${\varphi }_t\in L^2({\mathbb{R}}^3\to \mathbb{C})$ such that ${\lim }_{N\to \infty }{N}^{-1}<{\Psi }_t,{\mathcal{N}}^{{\varphi }_t}{\Psi }_t>=1$ and that ${\varphi }_t$ is -- dependent of the scaling of the range of the interaction -- solution of the Gross-Pitaevskii or Hartree equation. We shall also show that under additional decay conditions of ${\varphi }_t$ the limit can be taken uniform in $t<\infty$ and that convergence of the relative particle number implies convergence of the $k$-particle density matrices of ${\Psi }_t$.