Équations de champ moyen pour la dynamique quantique d'un grand nombre de particules  [ Mean field equations for the quantum dynamics of many particles ]
Séminaire Bourbaki : volume 2003/2004, exposés 924-937, Astérisque, no. 299 (2005), Talk no. 930, p. 147-164

The purpose of this talk is to describe how the Schrödinger evolution for the quantum N-body problem is approximated, as N tends to infinity, in a suitable regime, by a nonlinear evolution in three space dimensions. We shall discuss the case of bosons, which leads to the Schrödinger-Poisson equation, and the case of fermions, with its connections to the Hartree-Fock system.

L’objet de cet exposé est de montrer comment l’évolution de Schrödinger pour le problème à N corps quantique est approchée, lorsque N tend vers l’infini, dans un régime convenable, par une évolution non-linéaire en dimension trois d’espace. On traitera le cas des bosons, qui conduit à l’équation de Schrödinger-Poisson, et celui des fermions, qui débouche sur le système de Hartree-Fock.

Classification:  35Q40,  35Q55,  81Q05,  81V70
Keywords: quantum N-body problem, mean field equations, Hartree equation, Schrödinger-Poisson equation, Hartree-Fock system, Slater determinants, BBGKY hierarchy
@incollection{SB_2003-2004__46__147_0,
     author = {G\'erard, Patrick},
     title = {\'Equations de champ moyen pour la dynamique quantique d'un grand nombre de particules},
     booktitle = {S\'eminaire Bourbaki : volume 2003/2004, expos\'es 924-937},
     author = {Collectif},
     series = {Ast\'erisque},
     publisher = {Association des amis de Nicolas Bourbaki, Soci\'et\'e math\'ematique de France},
     address = {Paris},
     number = {299},
     year = {2005},
     note = {talk:930},
     pages = {147-164},
     zbl = {1079.35084},
     language = {fr},
     url = {http://www.numdam.org/item/SB_2003-2004__46__147_0}
}
Gérard, Patrick. Équations de champ moyen pour la dynamique quantique d'un grand nombre de particules, in Séminaire Bourbaki : volume 2003/2004, exposés 924-937, Astérisque, no. 299 (2005), Talk no. 930, pp. 147-164. http://www.numdam.org/item/SB_2003-2004__46__147_0/

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