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$% L. Baudouin, Existence and regularity of the solution of a time dependent Hartree-Fock equation cvoupled with a cvlassical nuclear dynamics, Rev. Mat. Complut. 18 (2005), no. 2, 285314.%$

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

Laboratoire de Mathématiques Appliquées
Université de Versailles Saint-Quentin
45 avenue des Etats Unis
78035 Versailles Cedex France.

Received: July 5, 2004.
Accepted: November 17, 2004.

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.