From Schrödinger to Hartree--Fock and Vlasov equations with Singular potentials

Jacky Chong, Laurent Lafleche, Chiara Saffirio

Published 2021-03-19Version 1

We obtain the combined mean-field and semiclassical limit from the $N$-body Schr\"odinger equation for Fermions in case of singular potentials. In order to obtain this result, we first prove the propagation of regularity uniformly in the Planck constant $h$ for the Hartree--Fock equation with singular pair interaction potentials of the form $|x-y|^{-a}$, including the Coulomb interaction. We then use these bounds to obtain quantitative bounds on the distance between solutions of the Schr\"odinger 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.