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

TL;DR

This paper introduces a novel method to derive the time-dependent Gross-Pitaevskii equation from microscopic quantum dynamics without requiring the interaction to be positive, under certain scaling and initial state assumptions.

Contribution

It extends the derivation of the Gross-Pitaevskii equation to cases with non-positive interactions by relaxing previous positivity constraints using a new approach.

Findings

Derivation valid for scaling parameter <1/3

Requires initial reduced density matrix to converge rapidly to a pure state

Method avoids BBGKY hierarchy in mean field derivation

Abstract

Using a new method it is possible to derive mean field equations from the microscopic $N$ body Schr\"odinger 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 $\beta<1/3$ and that we have to assume fast convergence of the reduced one particle marginal density matrix of the initial wave function $\mu^{\Psi_0}$ to a pure state $|\phi_0><\phi_0|$.