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On the Mean-Field Limit of Bosons with Coulomb Two-Body Interaction

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Abstract

In the mean-field limit the dynamics of a quantum Bose gas is described by a Hartree equation. We present a simple method for proving the convergence of the microscopic quantum dynamics to the Hartree dynamics when the number of particles becomes large and the strength of the two-body potential tends to 0 like the inverse of the particle number. Our method is applicable for a class of singular interaction potentials including the Coulomb potential. We prove and state our main result for the Heisenberg- picture dynamics of “observables”, thus avoiding the use of coherent states. Our formulation shows that the mean-field limit is a “semi-classical” limit.

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Authors and Affiliations

  1. Institute of Theoretical Physics, ETH Hönggerberg, CH-8093, Zürich, Switzerland

    Jürg Fröhlich, Antti Knowles & Simon Schwarz

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  1. Jürg Fröhlich
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  2. Antti Knowles
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  3. Simon Schwarz
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Correspondence to Antti Knowles.

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Communicated by H.-T. Yau

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Fröhlich, J., Knowles, A. & Schwarz, S. On the Mean-Field Limit of Bosons with Coulomb Two-Body Interaction. Commun. Math. Phys. 288, 1023–1059 (2009). https://doi.org/10.1007/s00220-009-0754-z

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  • DOI: https://doi.org/10.1007/s00220-009-0754-z

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