Derivation of the time dependent Gross–Pitaevskii equation with external fields

Abstract

Using a new method [19], it is possible to derive mean field equations from the microscopic N body Schrödinger evolution of interacting particles without using BBGKY hierarchies. This method also allows for error estimates and can be generalized to systems with external fields, of which both points are relevant from a physics perspective. Recently, this method was used to derive the Hartree equation for singular interactions [11] and the Gross–Pitaevskii equation without positivity condition on the interaction [17] where one had to restrict the scaling behavior of the interaction. Assuming positivity of the interaction, this paper deals with more general scalings including the so-called Gross–Pitaevskii scaling.