On the Time Dependent Gross Pitaevskii- and Hartree Equation

We are interested in solutions Ψt 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 N0 be the particle number operator with respect to some φ0 ∈ L (R → C). Assume that the relative particle number of the initial wave function N−1〈Ψ0,N 0Ψ0〉 converges to one as N → ∞. We shall show that we can find a φt ∈ L (R → C) such that limN→∞ N −1〈Ψt,N Ψt〉 = 1 and that φt 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 φt the limit can be taken uniform in t < ∞ and that convergence of the relative particle number implies convergence of the k-particle density matrices of Ψt.