Existence and Regularity of the Solution of a Time Dependent Hartree-Fock Equation Coupled with a Classical Nuclear Dynamics

Abstract

We study an Helium atom (composed of one nucleus and two electrons) sub mitted to a general time dependent electric field, modeled by the Hartree-Fock equation, whose solution is the wave function of the electrons, coupled with the classical Newtonian dynamics, for the position of the nucleus. We prove a result of existence and regularity for the Cauchy problem, where the main ingredients are a preliminary study of the regularity in a nonlinear Schr¨ nger equation odi with semi-group techniques and a Schauder fixed point theorem.