Derivation of the Schrödinger-Poisson equation from the quantum \(N\)-body problem. (English. Abridged French version) Zbl 1018.81009

Summary: We derive the time-dependent Schrödinger-Poisson equation as the weak coupling limit of the \(N\)-body linear Schrödinger equation with Coulomb potential.


81Q05 Closed and approximate solutions to the Schrödinger, Dirac, Klein-Gordon and other equations of quantum mechanics
82B44 Disordered systems (random Ising models, random Schrödinger operators, etc.) in equilibrium statistical mechanics
35J10 Schrödinger operator, Schrödinger equation
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[1] Bardos, C.; Golse, F.; Mauser, N.J., Weak coupling limit of the N-particle Schrödinger equation, Methods appl. anal., 7, 275-293, (2000) · Zbl 1003.81027
[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) · Zbl 1145.82300
[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., 35, 265-277, (1974)
[5] Ginibre, J.; Velo, G., The classical field limit of scattering theory for non-relativistic many-boson systems. I and II, Comm. math. phys., 66, 37-76, (1979), and 68 (1979) 45-68 · Zbl 0443.35067
[6] Ginibre, J.; Velo, G., On a class of nonlinear Schrödinger equations with nonlocal interactions, Math. Z., 170, 109-145, (1980)
[7] Kato, T., Fundamental properties of Hamiltonian operators of Schrödinger type, Trans. amer. math. soc., 70, 195-211, (1951) · Zbl 0044.42701
[8] Leray, J., Sur le mouvement d’un liquide visqueux emplissant l’espace, Acta math., 63, 183-248, (1934) · JFM 60.0726.05
[9] Nishida, T., A note on a theorem of Nirenberg, J. differential geom., 12, 629-633, (1977) · Zbl 0368.35007
[10] Spohn, H., Kinetic equations from Hamiltonian dynamics, Rev. mod. phys., 52, 3, 569-615, (1980)
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