Derivation of the Time Dependent Gross-Pitaevskii Equation Without Positivity Condition on the Interaction

Abstract

Using a new method (Pickl in A simple derivation of mean field limits for quantum systems, 2010) 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 (Erdös et al. in Commun. Pure Appl. Math. 59(12):1659-1741, 2006; Invent. Math. 167:515-614, 2007). The new method for dealing with mean field limits presented in Pickl (2010) 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 of the interaction and that we have to assume fast convergence of the reduced one particle marginal density matrix of the initial wave function μ Ψ0 to a pure state {pipe}φ 0 〉〈φ 0 {pipe}. © 2010 Springer Science+Business Media, LLC.