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arXiv · 0808.1178

On the Time Dependent Gross Pitaevskii- and Hartree Equation

Peter Pickl
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Abstract

We are interested in solutions $Ψ_t$ 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 $\mathcal{N}^{ϕ_0}$ be the particle number operator with respect to some $ϕ_0\in L^2(\mathbb{R}^3\to\mathbb{C})$. Assume that the relative particle number of the initial wave function $N^{-1}< Ψ_0,\mathcal{N}^{ϕ_0}Ψ_0>$ converges to one as $N\to\infty$. We shall show that we can find a $ϕ_t\in L^2(\mathbb{R}^3\to\mathbb{C})$ such that $\lim_{N\to\infty}N^{-1}< Ψ_t,\mathcal{N}^{ϕ_t}Ψ_t>=1$ and that $ϕ_t$ 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 $ϕ_t$ 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 $Ψ_t$.

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Peter Pickl. 2008-08-08. On the Time Dependent Gross Pitaevskii- and Hartree Equation. https://arxiv.org/abs/0808.1178

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