Abstract
In this paper we consider the nonlinear Hartree equation in presence of a given external potential, for an initial coherent state. Under suitable smoothness assumptions, we approximate the solution in terms of a time dependent coherent state, whose phase and amplitude can be determined by a classical flow. The error can be estimated in L 2 by \({C \sqrt {\varepsilon}}\) , \({\varepsilon}\) being the Planck constant. Finally we present a full formal asymptotic expansion.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
Athanassoulis, A., Paul, T., Pezzotti, F., Pulvirenti, M.: Strong semiclassical approximation of Wigner functions for the Hartree dynamics. arXiv:1009.0470v1 [math-ph]
Bardos C., Golse F., Mauser N.: Weak coupling limit of the N-particle Schrödinger equation. Methods Appl. Anal. 7, 275–293 (2000)
Bardos C., Erdős L., Golse F., Mauser N., Yau H.-T.: Derivation of the Schrödinger-Poisson equation from the quantum N-body problem. C. R. Acad. Sci. Paris, Ser I. 334, 515–520 (2002)
Belov V.V., Kondratieva M.F., Smirnova E.I.: Semiclassical soliton-type solutions of the Hartree equation. Doklady Math. 76(2), 775–779 (2007)
Carles R., Fermanian-Kammerer C.: Nonlinear coherent states and Eherenfest time for Schrödinger equation. Commun. Math. Phys. 301(2), 443–472 (2010)
Cazenave T., Weissler F.: The Cauchy problem for the nonlinear Schrödinger equation in H 1. Manuscr. Math. 61, 477–494 (1988)
Erdős L., Yau H.-T.: Derivation of the nonlinear Schrödinger equation from a many body Coulomb system. Adv. Theor. Math. Phys. 5(6), 1169 (2001)
Ginibre J., Velo G.: The classical field limit of scattering theory for non-relativistic many-boson systems: I. Commun. Math. Phys. 66, 37–76 (1979)
Ginibre J., Velo G.: The classical field limit of scattering theory for non-relativistic many-boson systems: II. Commun. Math. Phys. 68, 45–68 (1979)
Ginibre J., Velo G.: On a class of non linear Schrödinger equations with non local interactions. Math. Z. 170(2), 109–136 (1980)
Hagedorn G.A.: Semiclassical quantum mechanics. I. The \({\hbar\to 0}\) limit for coherent states. Commun. Math. Phys. 71(1), 77 (1980)
Hagedorn G.A.: Raising and lowering operators for semiclassical wave packets. Ann. Phys. 269, 77–104 (1998)
Hepp K.: The classical limit for quantum mechanical correlation functions. Commun. Math. Phys. 35, 265–277 (1974)
Knowles A., Pickl P.: Mean-field dynamics: singular potentials and rate of convergence . Commun. Math. Phys. 298(1), 101–138 (2010)
Lions P.-L., Paul T.: Sur les mesures de Wigner. Rev. Mat. Iberoamericana 9(3), 553–618 (1993)
Lisok A., Trifnov A.Yu., Shapovalov A.V.: The evolution operator of the Hartree-type equation with a quadratic potential. J. Phys. A 37, 4535 (2004)
Paul, T.: Semiclassical methods with an emphasis on coherent states. In: Simon, B., Rauch, J. (eds.) Tutorial Lectures, Proceedings of the Conference “Quasiclassical methods”, IMA Series. Springer, Berlin (1997)
Paul, T.: Échelles de temps pour l’évolution quantique à petite constante de Planck. Séminaire X-EDP 2007–2008. Publications de l’École Polytechnique (2008)
Pezzotti, F.: Mean-field limit and semiclassical approximation for quantum particle systems. PhD thesis. Rendiconti di Matematica e delle sue Applicazioni 29 (2009)
Pezzotti F., Pulvirenti M.: Mean-field limit and Semiclassical Expansion of a Quantum Particle System. Ann. H. Poincaré 10(1), 145–187 (2009)
Reed M., Simon B.: Methods of Modern Mathematical Physics, vol. II. Academic Press, New York (1975)
Spohn H.: Kinetic equations from Hamiltonian dynamics: Markovian limits. Rev. Modern Phys. 53(3), 569–615 (1980)
Author information
Authors and Affiliations
Corresponding author
Additional information
Communicated by Rafael D. Benguria.
Rights and permissions
About this article
Cite this article
Athanassoulis, A., Paul, T., Pezzotti, F. et al. Semiclassical Propagation of Coherent States for the Hartree Equation. Ann. Henri Poincaré 12, 1613–1634 (2011). https://doi.org/10.1007/s00023-011-0115-2
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00023-011-0115-2