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

Jacky Chong, Laurent Lafleche, Chiara Saffirio

Jacky Chong, Laurent Lafleche, Chiara Saffirio

**19/3/21**Published in : arXiv:2103.10946

We obtain the combined mean-field and semiclassical limit from the N-body Schrödinger 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 Schrödinger 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.