no. 6
Derivation of the Schrödinger–Poisson equation from the quantum 𝐍-body problem
[Justification de l'équation de Schrödinger–Poisson à partir du problème quantique à N corps]
Bardos, Claude1 ; Erdös, Laszlo2 ; Golse, François1 ; Mauser, Norbert3 ; Yau, Horng-Tzer4
1 Université Paris 6, Laboratoire d'analyse numérique, 175, rue du Chevaleret, 75013 Paris, France
2 School of Mathematics, Georgia Institute of Technology, Atlanta, GA 30332, USA
3 Wolfgang Pauli Institute, c/o Inst. f. das Mathematik, Universität Wien, Strudlhofgasse 4, A-1090 Wien, Austria
4 Courant Institute of Mathematical Sciences, 251 Mercer Street, New York, NY 10012, USA
Comptes Rendus. Mathématique, Tome 334 (2002) no. 6, pp. 515-520.

On établit la validité de l'équation de Schrödinger–Poisson en régime instationnaire comme limite à couplage faible de l'équation de Schrödinger linéaire à N corps avec potentiel de Coulomb.

We derive the time-dependent Schrödinger–Poisson equation as the weak coupling limit of the N-body linear Schrödinger equation with Coulomb potential.

Reçu le :
Accepté le :
Publié le :
DOI : 10.1016/S1631-073X(02)02253-7
Bardos, Claude 1 ; Erdös, Laszlo 2 ; Golse, François 1 ; Mauser, Norbert 3 ; Yau, Horng-Tzer 4

1 Université Paris 6, Laboratoire d'analyse numérique, 175, rue du Chevaleret, 75013 Paris, France
2 School of Mathematics, Georgia Institute of Technology, Atlanta, GA 30332, USA
3 Wolfgang Pauli Institute, c/o Inst. f. das Mathematik, Universität Wien, Strudlhofgasse 4, A-1090 Wien, Austria
4 Courant Institute of Mathematical Sciences, 251 Mercer Street, New York, NY 10012, USA 
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     title = {Derivation of the {Schr\"odinger{\textendash}Poisson} equation from the quantum $ \mathbf{N}$-body problem},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {515--520},
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Bardos, Claude; Erdös, Laszlo; Golse, François; Mauser, Norbert; Yau, Horng-Tzer. Derivation of the Schrödinger–Poisson equation from the quantum $ \mathbf{N}$-body problem. Comptes Rendus. Mathématique, Tome 334 (2002) no. 6, pp. 515-520. doi : 10.1016/S1631-073X(02)02253-7. http://archive.numdam.org/articles/10.1016/S1631-073X(02)02253-7/

[1] Bardos, C.; Golse, F.; Mauser, N.J. Weak coupling limit of the N-particle Schrödinger equation, Methods Appl. Anal., Volume 7 (2000), pp. 275-293

[2] C. Bardos, F. Golse, A. Gottlieb, N.J. Mauser, On the derivation of nonlinear Schrödinger and Vlasov equations, Proceedings of the I.M.A., Springer-Verlag (to appear)

[3] L. Erdös, H.-T. Yau, Derivation of the nonlinear Schrödinger equation with Coulomb potential, Preprint

[4] Hepp, K. The classical limit for quantum mechanical correlation functions, Comm. Math. Phys., Volume 35 (1974), pp. 265-277

[5] Ginibre, J.; Velo, G. The classical field limit of scattering theory for non-relativistic many-boson systems. I and II, Comm. Math. Phys., Volume 66 (1979), pp. 37-76 and 68 (1979) 45–68

[6] Ginibre, J.; Velo, G. On a class of nonlinear Schrödinger equations with nonlocal interactions, Math. Z., Volume 170 (1980), pp. 109-145

[7] Kato, T. Fundamental properties of Hamiltonian operators of Schrödinger type, Trans. Amer. Math. Soc., Volume 70 (1951), pp. 195-211

[8] Leray, J. Sur le mouvement d'un liquide visqueux emplissant l'espace, Acta Math., Volume 63 (1934), pp. 183-248

[9] Nishida, T. A note on a theorem of Nirenberg, J. Differential Geom., Volume 12 (1977), pp. 629-633

[10] Spohn, H. Kinetic equations from hamiltonian dynamics, Rev. Mod. Phys., Volume 52 (1980) no. 3, pp. 569-615

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