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Strong semiclassical limits from Hartree and Hartree–Fock to Vlasov–Poisson equations

Laurent Lafleche and Chiara Saffirio

Vol. 16 (2023), No. 4, 891–926
Abstract

We consider the semiclassical limit from the Hartree to the Vlasov equation with general singular interaction potential including the Coulomb and gravitational interactions, and we prove explicit bounds in the strong topologies of Schatten norms. Moreover, in the case of fermions, we provide estimates on the size of the exchange term in the Hartree–Fock equation and also obtain a rate of convergence for the semiclassical limit from the Hartree–Fock to the Vlasov equation in Schatten norms. Our results hold for general initial data in some Sobolev space and any fixed time interval.

Keywords
Hartree equation, Hartree–Fock equation, Vlasov equation, Coulomb interaction, gravitational interaction, semiclassical limit
Mathematical Subject Classification
Primary: 35Q41, 35Q55, 82C10
Secondary: 35Q83, 82C05
Milestones
Received: 4 May 2020
Revised: 20 September 2021
Accepted: 19 October 2021
Published: 15 June 2023
Authors
Laurent Lafleche
Department of Mathematics
The University of Texas at Austin
Austin, TX
United States
Chiara Saffirio
Department of Mathematics and Computer Science
University of Basel
Basel
Switzerland

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