Vol. 6, No. 7, 2013

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ISSN: 1948-206X (e-only)
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The semiclassical limit of the time dependent Hartree–Fock equation: The Weyl symbol of the solution

Laurent Amour, Mohamed Khodja and Jean Nourrigat

Vol. 6 (2013), No. 7, 1649–1674
Abstract

For a family of solutions to the time dependent Hartree–Fock equation, depending on the semiclassical parameter h, we prove that if at the initial time the Weyl symbol of the solution is in L1(2n) as well as all its derivatives, then this property is true for all time, and we give an asymptotic expansion in powers of h of this Weyl symbol. The main term of the asymptotic expansion is a solution to the Vlasov equation, and the error term is estimated in the norm of L1(2n).

Keywords
time dependent Hartree–Fock equation, Vlasov equation, semiclassical analysis, Egorov theorem, pseudodifferential operators
Mathematical Subject Classification 2010
Primary: 35S05, 81Q20, 82C10
Milestones
Received: 8 June 2012
Revised: 26 November 2012
Accepted: 19 January 2013
Published: 27 December 2013
Authors
Laurent Amour
Laboratoire de Mathématiques de Reims
Université de Reims Champagne-Ardenne
EA 4535, FR CNRS-3399
51687 Reims
France
Mohamed Khodja
Laboratoire de Mathématiques de Reims
Université de Reims Champagne-Ardenne
EA 4535, FR CNRS-3399
51687 Reims
France
Jean Nourrigat
Laboratoire de Mathématiques de Reims
Université de Reims Champagne-Ardenne
EA 4535, FR CNRS-3399
51687 Reims
France