From many-body quantum dynamics to the Hartree-Fock and Vlasov equations with singular potentials

We obtain the combined mean-field and semiclassical limit from the $N$-body Schr"{o}dinger equation for fermions interacting via singular potentials. To obtain the result, we first prove the uniformity in Planck’s constant $h$ propagation of regularity for solutions to the Hartree$\text{unicode}x2013$Fock equation with singular pair interaction potentials of the form $\pm |x-y|^{-a}$, including the Coulomb and gravitational interactions. In the context of mixed states, we use these regularity properties to obtain quantitative estimates on the distance between solutions to the Schr"{o}dinger equation and solutions to the Hartree$\text{unicode}x2013$Fock and Vlasov equations in Schatten norms. For $a\in (0,1\mathrm{/}2)$, we obtain local-in-time results when $N^{-1\mathrm{/}2}\ll h\le N^{-1\mathrm{/}3}$. In particular, it leads to the derivation of the Vlasov equation with singular potentials. For $a\in [1\mathrm{/}2,1]$, our results hold only on a small time scale, or with an $N$-dependent cutoff.