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 ${\omega }_N$ 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 ${\omega }_N$. Our result holds for all (semiclassical) times, and gives effective bounds on the rate of the convergence towards the Hartree-Fock dynamics.

42 pages; some references added; paragraphs on semiclassical structure and on construction of the Bogoliubov transformation added; some clarifications

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• Rigorous Derivation of the Gross-Pitaevskii Equation
• On the Mean-Field Limit of Bosons with Coulomb Two-Body Interaction
• Critical Temperature and Energy Gap for the BCS Equation
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