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

Original · EN

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 μΨ⁰ to a pure state |ϕ₀><ϕ₀|.