# Strong Semiclassical Approximation of Wigner Functions for the Hartree Dynamics

Abstract : We consider the Wigner equation corresponding to a nonlinear Schroedinger evolution of the Hartree type in the semiclassical limit $\hbar\to 0$. Under appropriate assumptions on the initial data and the interaction potential, we show that the Wigner function is close in $L^2$ to its weak limit, the solution of the corresponding Vlasov equation. The strong approximation allows the construction of semiclassical operator-valued observables, approximating their quantum counterparts in Hilbert-Schmidt topology. The proof makes use of a pointwise-positivity manipulation, which seems necessary in working with the $L^2$ norm and the precise form of the nonlinearity. We employ the Husimi function as a pivot between the classical probability density and the Wigner function, which -- as it is well known -- is not pointwise positive in general.
Type de document :
Pré-publication, Document de travail
24 pages. 2010
Domaine :

https://hal.inria.fr/inria-00528983
Contributeur : Agissilaos Athanassoulis <>
Soumis le : samedi 23 octobre 2010 - 16:06:29
Dernière modification le : mercredi 27 juillet 2016 - 14:48:48

### Identifiants

• HAL Id : inria-00528983, version 1
• ARXIV : 1009.0470

### Citation

A. Athanassoulis, T. Paul, F. Pezzotti, M. Pulvirenti. Strong Semiclassical Approximation of Wigner Functions for the Hartree Dynamics. 24 pages. 2010. 〈inria-00528983〉

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