Strong Semiclassical Approximation of Wigner Functions for the Hartree Dynamics

TL;DR

This paper develops a strong semiclassical approximation for Wigner functions in Hartree dynamics, enabling better quantum-classical correspondence and operator approximation in the semiclassical limit.

Contribution

It introduces a novel strong approximation method for Wigner functions in Hartree dynamics, utilizing Husimi functions and positivity techniques.

Findings

Wigner function closely approximates the Vlasov solution in L^2 norm.

Constructs semiclassical operator-valued observables with Hilbert-Schmidt accuracy.

Employs Husimi functions to bridge quantum and classical descriptions.

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.