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

Lucie BAUDOUIN

Université de Versailles Saint-Quentin

45 avenue des Etats Unis

78035 Versailles Cedex — France.

Accepted: November 17, 2004.

ABSTRACT

We study an Helium atom (composed of one nucleus and two electrons) submitted 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 Schrödinger equation with semi-group techniques and a Schauder fixed point theorem.

Key words: Hartree-Fock equation, classical dynamics, regularity, existence.

2000 Mathematics Subject Classification: 35Q40, 35Q55, 34A12.