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Mean–Field Evolution of Fermionic Systems

Abstract

The mean field limit for systems of many fermions is naturally coupled with a semiclassical limit. This makes the analysis of the mean field regime much more involved, compared with bosonic systems. In this paper, we study the dynamics of initial data close to a Slater determinant, whose reduced one-particle density is an orthogonal projection ω N with the appropriate semiclassical structure. Assuming some regularity of the interaction potential, we show that the evolution of such an initial data remains close to a Slater determinant, with reduced one-particle density given by the solution of the Hartree–Fock equation with initial data ω N . Our result holds for all (semiclassical) times, and gives effective bounds on the rate of the convergence towards the Hartree–Fock dynamics.

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Correspondence to Benjamin Schlein.

Additional information

Communicated by H. Spohn

M. Porta supported by ERC Grant MAQD 240518.

B. Schlein partially supported by ERC Grant MAQD 240518.

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Benedikter, N., Porta, M. & Schlein, B. Mean–Field Evolution of Fermionic Systems. Commun. Math. Phys. 331, 1087–1131 (2014). https://doi.org/10.1007/s00220-014-2031-z

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  • DOI: https://doi.org/10.1007/s00220-014-2031-z

Keywords

  • Fermionic System
  • Slater Determinant
  • Hartree Equation
  • Bogoliubov Transformation
  • Canonical Anticommutation Relation