On the Time Dependent Gross Pitaevskii- and Hartree Equation

Abstract

We are interested in solutions $\Psi_t$ of the Schr\"odinger 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 $\mathcal{N}^{\phi_0}$ be the particle number operator with respect to some $\phi_0\in L^2(\mathbb{R}^3\to\mathbb{C})$. Assume that the relative particle number of the initial wave function $N^{-1}< \Psi_0,\mathcal{N}^{\phi_0}\Psi_0>$ converges to one as $N\to\infty$. We shall show that we can find a $\phi_t\in L^2(\mathbb{R}^3\to\mathbb{C})$ such that $\lim_{N\to\infty}N^{-1}< \Psi_t,\mathcal{N}^{\phi_t}\Psi_t>=1$ and that $\phi_t$ 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 $\phi_t$ 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 $\Psi_t$.