#### Vol. 6, No. 7, 2013

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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 ${L}^{1}\left({ℝ}^{2n}\right)$ 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 ${L}^{1}\left({ℝ}^{2n}\right)$.

##### Keywords
time dependent Hartree–Fock equation, Vlasov equation, semiclassical analysis, Egorov theorem, pseudodifferential operators
##### Mathematical Subject Classification 2010
Primary: 35S05, 81Q20, 82C10