Strong Semiclassical Approximation of Wigner Functions for the Hartree Dynamics

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.