# Strong semiclassical approximation of Wigner functions for the Hartree dynamics

### Agissilaos Athanassoulis

University of Cambridge, UK### Thierry Paul

Ecole Polytechnique, Palaiseau, France### Federica Pezzotti

Universidad del Pais Vasco, Bilbao, Spain### Mario Pulvirenti

Università di Roma La Sapienza, Italy

## Abstract

We consider the Wigner equation corresponding to a nonlinear Schrödinger evolution of the Hartree type in the semiclassical limit $h \rightarrow 0$.

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.

## Cite this article

Agissilaos Athanassoulis, Thierry Paul, Federica Pezzotti, Mario Pulvirenti, Strong semiclassical approximation of Wigner functions for the Hartree dynamics. Atti Accad. Naz. Lincei Cl. Sci. Fis. Mat. Natur. 22 (2011), no. 4, pp. 525–552

DOI 10.4171/RLM/613