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Mean-Field Dynamics: Singular Potentials and Rate of Convergence

Abstract

We consider the time evolution of a system of N identical bosons whose interaction potential is rescaled by N −1. We choose the initial wave function to describe a condensate in which all particles are in the same one-particle state. It is well known that in the mean-field limit N → ∞ the quantum N-body dynamics is governed by the nonlinear Hartree equation. Using a nonperturbative method, we extend previous results on the mean-field limit in two directions. First, we allow a large class of singular interaction potentials as well as strong, possibly time-dependent external potentials. Second, we derive bounds on the rate of convergence of the quantum N-body dynamics to the Hartree dynamics.

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Correspondence to Antti Knowles.

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Communicated by H.-T. Yau

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Knowles, A., Pickl, P. Mean-Field Dynamics: Singular Potentials and Rate of Convergence. Commun. Math. Phys. 298, 101–138 (2010). https://doi.org/10.1007/s00220-010-1010-2

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  • DOI: https://doi.org/10.1007/s00220-010-1010-2

Keywords

  • Singular Potential
  • Trace Class Operator
  • Hartree Equation
  • BBGKY Hierarchy
  • Continuous Nonnegative Function