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Quantum Fluctuations and Rate of Convergence Towards Mean Field Dynamics

Abstract

The nonlinear Hartree equation describes the macroscopic dynamics of initially factorized N-boson states, in the limit of large N. In this paper we provide estimates on the rate of convergence of the microscopic quantum mechanical evolution towards the limiting Hartree dynamics. More precisely, we prove bounds on the difference between the one-particle density associated with the solution of the N-body Schrödinger equation and the orthogonal projection onto the solution of the Hartree equation.

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Correspondence to Benjamin Schlein.

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Communicated by H.-T. Yau

Partially supported by NSF grant DMS-0702270.

On leave from the University of Cambridge. Supported by a Sofja Kovalevskaja Award of the Humboldt Foundation. Current address: University of Cambridge, Centre for Mathematical Sciences, DPMMS, Wilberforce Rd, Cambridge CB3 0WB, UK.

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Rodnianski, I., Schlein, B. Quantum Fluctuations and Rate of Convergence Towards Mean Field Dynamics. Commun. Math. Phys. 291, 31–61 (2009). https://doi.org/10.1007/s00220-009-0867-4

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  • DOI: https://doi.org/10.1007/s00220-009-0867-4

Keywords

  • Coherent State
  • Quantum Fluctuation
  • Field Dynamics
  • Singular Potential
  • Canonical Commutation Relation