Hartree corrections in a mean-field limit for fermions with Coulomb interaction**This article belongs to the special issue: Emerging Talents, which features invited work from the best early-career researchers working within the scope of J. Phys. A. This project is part of the Journal of Physics series’ 50th anniversary celebrations in 2017. Sören Petrat was selected by the Editorial Board of

Abstract

We consider the many-body dynamics of fermions with Coulomb interaction in a mean-field scaling limit where the kinetic and potential energy are of the same order for large particle numbers. In the considered limit the spatial variation of the mean-field is small. We prove two results about this scaling limit. First, due to the small variation, i.e., small forces, we show that the many-body dynamics can be approximated by the free dynamics with an appropriate phase factor with the conjectured optimal error term. Second, we show that the Hartree dynamics gives a better approximation with a smaller error term. In this sense, assuming that the error term in the first result is optimal, we derive the Hartree equations from the many-body dynamics with Coulomb interaction in a mean-field scaling limit.