We consider the combined mean-field and semiclassical limit for a system of the N interacting Fermions in the case of singular potentials. We prove the uniformly in the Planck constant h propagation of regularity for the Hartree--Fock equation with singular pair interaction potentials of the form $|x-y|^{-a}$, including the Coulomb interaction. Using these estimates, we obtain quantitative bounds on the distance between solutions of the Schrodinger equation and solutions of Hartree--Fock and Vlasov equations in Schatten norms. For $a \in (0, 1/2)$, we obtain local-in-time results when $N^{-1/2} \ll h \leq N^{-1/3}$. In particular, it leads to the derivation of the Vlasov equation with singular potentials. For $a\in (1/2,1]$, our results hold only on a small time scale $t\sim h^{a-1/2}$, or with a $N$ dependent cutoff. This is a joint work with Laurent Lafleche and  Chiara Saffirio.

Abstract

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