Derivation of the Time Dependent Gross Pitaevskii Equation with External Fields

TL;DR

This paper introduces a novel method to derive the time-dependent Gross Pitaevskii equation from microscopic quantum dynamics, accommodating general interaction scalings under positivity assumptions, bypassing traditional BBGKY hierarchy techniques.

Contribution

It extends a new derivation method to more general interaction scalings for the Gross Pitaevskii equation, avoiding BBGKY hierarchies and positivity restrictions.

Findings

Derived the time-dependent Gross Pitaevskii equation for general scalings

Established the method's applicability without BBGKY hierarchies

Confirmed the positivity condition on interactions

Abstract

Using a new method [9] it is possible to derive mean field equations from the microscopic N-body Schrodinger evolution of interacting particles without using BBGKY hierarchies. Recently this method was used to derive the Hartree equation for singular interactions [4] and the Gross Pitaevskii equation without positivity condition on the interaction [10] where one had to restrict the scaling behaviour of the interaction. In this paper more general scalings shall be considered assuming positivity of the interaction.