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

Abstract: 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$\unicode{x2013}$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$\unicode{x2013}$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, or with an $N$-dependent cutoff.