Strong Semiclassical Approximation of Wigner Functions for the Hartree Dynamics

Details Strong Semiclassical Approximation of Wigner Functions for the Hartree Dynamics Strong Semiclassical Approximation of Wigner Functions for the Hartree Dynamics

2010-09-02

arxiv.org/abs/1009.0470v1

Physics

Mathematical Physics

24 pages

Scientific paper

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.

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