The purpose of this talk is to describe how the Schrödinger evolution for the quantum -body problem is approximated, as 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 corps quantique est approchée, lorsque 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.
Mot clés : problème à $N$ corps quantique, équations de champ moyen, équation de Hartree, équation de Schrödinger-Poisson, système de Hartree-Fock, déterminants de Slater, hiérarchie BBGKY
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}, series = {Ast\'erisque}, note = {talk:930}, pages = {147--164}, publisher = {Association des amis de Nicolas Bourbaki, Soci\'et\'e math\'ematique de France}, address = {Paris}, number = {299}, year = {2005}, zbl = {1079.35084}, language = {fr}, url = {http://archive.numdam.org/item/SB_2003-2004__46__147_0/} }
TY - CHAP AU - Gérard, Patrick TI - Équations de champ moyen pour la dynamique quantique d'un grand nombre de particules BT - Séminaire Bourbaki : volume 2003/2004, exposés 924-937 AU - Collectif T3 - Astérisque N1 - talk:930 PY - 2005 SP - 147 EP - 164 IS - 299 PB - Association des amis de Nicolas Bourbaki, Société mathématique de France PP - Paris UR - http://archive.numdam.org/item/SB_2003-2004__46__147_0/ LA - fr ID - SB_2003-2004__46__147_0 ER -
%0 Book Section %A Gérard, Patrick %T Équations de champ moyen pour la dynamique quantique d'un grand nombre de particules %B Séminaire Bourbaki : volume 2003/2004, exposés 924-937 %A Collectif %S Astérisque %Z talk:930 %D 2005 %P 147-164 %N 299 %I Association des amis de Nicolas Bourbaki, Société mathématique de France %C Paris %U http://archive.numdam.org/item/SB_2003-2004__46__147_0/ %G fr %F 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://archive.numdam.org/item/SB_2003-2004__46__147_0/
[1] “Remarks on the abstract form of nonlinear Cauchy-Kovalevsky theorems”, Comm. Partial Differential Equations 2 (1977), p. 1151-1162. | MR | Zbl
& -[2] “Derivation of the Schrödinger-Poisson equation from the quantum -body problem”, C. R. Acad. Sci. Paris Sér. I Math. 334 (2002), p. 515-520. | MR | Zbl
, , , & -[3] “Mean field dynamics of fermions and the time-dependent Hartree-Fock equation”, J. Math. Pures Appl. 82 (2003), p. 665-683. | MR | Zbl
, , & -[4] -, “Accuracy of the time-dependent Hartree-Fock approximation for uncorrelated initial states”, J. Statist. Phys. 115 (2004), à paraître. | MR | Zbl
[5] -, “Derivation of the time-dependent Hartree-Fock equation with Coulomb potential”, manuscrit, 2004.
[6] “Weak coupling limit of the -particle Schrödinger equation”, Methods Appl. Anal. 7 (2000), p. 275-293. | MR | Zbl
, & -[7] “On the Hartree-Fock time-dependent problem”, Comm. Math. Phys. 49 (1976), p. 25-33. | MR | Zbl
, & -[8] Mathematical theory of thermodynamic limits : Thomas-Fermi type models, Oxford Mathematical Monographs, Clarendon Press, 1998. | MR | Zbl
, & -[9] Semilinear Schrödinger Equations, Courant Lecture Notes in Mathematics, vol. 10, American Mathematical Society, 2003. | MR | Zbl
-[10] “Global existence of solutions to the Cauchy problem for time-dependent Hartree-Fock equations”, J. Math. Phys. 16 (1975), p. 1122-1130. | MR | Zbl
& -[11] “Note on exchange phenomena in the Thomas atom”, Math. Proc. Cambridge Philos. Soc. 26 (1930), p. 376-385. | JFM
-[12] “Derivation of the nonlinear Schrödinger equation from a many body Coulomb system”, Adv. Theor. Math. Phys. 5 (2001), p. 1169-1205. | MR | Zbl
& -[13] “Näherungsmethode zur Lösing des quantenmechanischen Mehrkörperproblems”, Z. Phys. 61 (1930), p. 126-148. | JFM
-[14] “The classical field limit of scattering theory for non-relativistic many-bosons systems, I-II”, Comm. Math. Phys. 66 (1979), p. 37-76, 68 (1979), p. 45-68. | MR | Zbl
& -[15] -, “On a class of nonlinear Schrödinger equations with nonlocal interactions”, Math. Z. 170 (1980), p. 109-145. | MR
[16] “The mean-field limit for the dynamics of large particle systems”, in Actes des Journées “Équations aux dérivées partielles”(Forges-les-Eaux, 2-6 juin 2003), GDR 2434 du CNRS, Version électronique disponible sur le site : http://www. math.sciences.univ-nantes.fr/edpa. | Numdam | MR | Zbl
-[17] “The wave mechanics of an atom with a non-Coulomb central field. Part I. Theory and methods”, Math. Proc. Cambridge Philos. Soc. 24 (1928), p. 89-132. | JFM
-[18] “The classical limit for quantum mechanical correlation functions”, Comm. Math. Phys. 35 (1974), p. 265-277. | MR
-[19] “Fundamental properties of Hamiltonian operators of Schrödinger type”, Trans. Amer. Math. Soc. 70 (1951), p. 195-201. | MR | Zbl
-[20] “The stability of matter : from atoms to stars”, Bull. Amer. Math. Soc. (N.S.) 22 (1990), p. 1-49. | MR | Zbl
-[21] “Inequalities for the moments of the eigenvalues of the Schrödinger Hamiltonian and their relation to Sobolev inequalities”, in Studies in Mathematical Physics, Essays in Honor of Valentine Bargmann (E.H. Lieb, B. Simon & W.E. Thirring, éds.), Princeton University Press, 1976. | Zbl
& -[22] “On an abstract form of the nonlinear Cauchy-Kowalewski theorem”, J. Differential Geom. 6 (1972), p. 561-576. | MR | Zbl
-[23] “A note on a theorem of Nirenberg”, J. Differential Geom. 12 (1977), p. 629-633. | MR | Zbl
-[24] “A nonlinear Cauchy problem in a scale of Banach spaces”, Dokl. Akad. Nauk SSSR 200 (1971), no. 4, traduction anglaise dans Sov. Math. Dokl. 12 (1971), p. 1497-1502. | Zbl
-[25] Methods of Modern Mathematical Physics, vol. I-IV, Academic Press, 1978. | MR | Zbl
& -[26] “A note on Hartree's method”, Phys. Rev. 35 (1930), p. 210-211.
-[27] “Kinetic equations from Hamiltonian dynamics : Markovian limits”, Rev. Modern Phys. 53 (1980), p. 569-615. | MR | Zbl
-[28] “-solutions for nonlinear Schrödinger equations and nonlinear groups”, Funkcial. Ekvac. 30 (1987), p. 115-125. | MR | Zbl
-[29] Gruppentheorie und Quantenmechanik, 1928, traduction et deuxième édition : The Theory of Groups and Quantum Mechanics, Dover, 1950. | JFM
-