Strong semiclassical approximation of Wigner functions for the Hartree dynamics

We consider the Wigner equation corresponding to a nonlinear Schrodinger evolution of the Hartree type in the semiclassical limit h -> 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.