Abstract
We present and discuss derivations of nonlinear 1-particle equations from linear N-particle Schrödinger equations with pair interaction in the time dependent case. We regard both the “classical” limit of vanishing Planck constant ħ → 0 which leads to Vlasov type equations and the “weak coupling” limit 1/N → 0 which leads to nonlinear 1 particle equations.
We use an approach to weak coupling limits where the so-called “finite Schrödinger hierarchy” and the limiting “(infinite) Schrödinger hierarchy” play a central role. Convergence of solutions of the first to solutions of the second is established using “physically relevant” estimates (L 2 and energy conservation) under very general assumptions on the interaction potential, including in particular the Coulomb potential. The goal of this work is to give an overview of the existing results, including some minor improvements, and clearly state the open problems.
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Refrences
C. Bardos, F. Golse, and N.J. Mauser, “Weak coupling limit of the N-particle Schrödinger equation”, to appear in Mathematical Analysis and Applications (2000).
O. Bokanowski and N.J. Mauser “Local approximation for the Hartree-Fock exchange potential: a deformation approach” Math. Models Methods Appl. Sci., vol. 9, No. 6, 941–961 (1999).
W. Braun and K. Hepp, “The Vlasov Dynamics and its fluctuation in the 1/N limit of interacting particles”, Comm. Math. Phys. 56 (1977), 101–113.
T. Cazenave and A. Haraux, Introduction aux problèmes d’évolution semilinéaires, Ellipses, Paris (1990).
C. Cercignani, Ludwig Boltzmann, Oxford Univ. Press (1998).
P. Gérard, P.A. Markowich, N.J. Mauser, and F. Poupaud, Homogenization Limits and Wigner Transforms, Comm. Pure and Appl. Math. 50 (1997), 321–377.
J. Ginibre and G. Velo, “On a class of nonlinear Schrödinger equations with nonlocal interaction”, Math. Zeitschr. 169 (1980), 109–145.
L. Hör mander, The analysis of linear partial differential operators, vol. 1, Springer Verlag, Berlin-New York (1983).
T. Kato, Perturbation Theory for Linear Operators, Springer-Verlag, (1966).
T. Kato, “Fundamental properties of Hamiltonian Operators of Schrödinger type “, Trans. Amer. Math. Soc. 70 (1951), 195–211.
L. Landau and E. Lifshitz, Quantum Mechanics, Course in Theoretical Physics, vol. 3, Pergamon Press, Oxford-New York-Toronto (1974).
J. Leray, “Sur le mouvement d’un liquide visqueux emplissant l’espace”, Acta Math. 63 (1934), 183–248.
E. Lieb and H.-T. Yau, “The Chandrasekhar Theory of Stellar Collapse as the Limit of Quantum Mechanic”, Commun. Math. Phys. 112 (1987), 147–174.
P.L. Lions and T. Paul, “Sur les mesures de Wigner”, Revista Math. Iberoamericana 9 (1993), 553–617.
P. Markowich and N. Mauser, “The Classical Limit of a Self-consistent Quantum-Vlasov Equation in 3-D”, Math. Mod. and Meth. in the Appl. Sci. 9 (1993), 109–124.
N.J. Mauser, “The Schrödinger-Poisson-Xα model”, to appear in Appl. Math. Lett. (2000)
H. Narnhofer and G.L. Sewell, “Vlasov Hydrodynamics of a Quantum Mechanical Model”, Commun. Math. Phys. 79 (1981), 9–24.
L. Nirenberg, “An abstract form of the nonlinear Cauchy-Kowalewski Theorem”, J. of Diff. Geom. 6 (1972), 561–576.
T. Nishida, “A note on a theorem of Nirenberg”, J. of Diff. Geom. 12 (1978), 629–633.
M. Reed and B. Simon, Methods of Modern Mathematical Physics, vol. 2, Academic Press, 4th ed. (1987).
G. Sewell, “Quantum Plasma Model with Hydrodynamical Phase Transition”, Helv. Phys. Acta 67 (1994), 4–19.
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Bardos, C., Golse, F., Gottlieb, A., Mauser, N.J. (2004). On the Derivation of Nonlinear Schrödinger and Vlasov Equations. In: Abdallah, N.B., et al. Dispersive Transport Equations and Multiscale Models. The IMA Volumes in Mathematics and its Applications, vol 136. Springer, New York, NY. https://doi.org/10.1007/978-1-4419-8935-2_1
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DOI: https://doi.org/10.1007/978-1-4419-8935-2_1
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