Approximative composition of Wick symbols and applications to the time dependent Hartree–Fock equation

In this paper, we derive a semiclassical asymptotic expansion of the Wick symbol, for the product of two operators, when the first one is a Weyl pseudodifferential operator in the class of Calderón–Vaillancourt and the second one is trace class in L²(Rⁿ). This result is extended in the direction of Schatten classes. We also report a corresponding result in the context of anti-Wick operators. The Wick expansion is used to derive an equation satisfied, up to an arbitrary small error term, by the Wick symbol of density operators governed by the time dependent Hartree–Fock (TDHF) equation. Still in the situation of the TDHF dynamics, we prove the finiteness of the Ehrenfest time.