Derivation of the time dependent Gross-Pitaevskii equation without positivity condition on the interaction

Using a new method it is possible to derive mean field equations from the microscopic $N$ body Schrödinger evolution of interacting particles without using BBGKY hierarchies. In this paper we wish to analyze scalings which lead to the Gross-Pitaevskii equation which is usually derived assuming positivity of the interaction. The new method for dealing with mean field limits presented in [6] allows us to relax this condition. The price we have to pay for this relaxation is however that we have to restrict the scaling behavior to $β<1/3$ and that we have to assume fast convergence of the reduced one particle marginal density matrix of the initial wave function $μ^{Ψ_0}$ to a pure state $|ϕ_0><ϕ_0|$.

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