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

Abstract

In a combined mean-field and semiclassical regime, we consider the time evolution of $N$ fermions interacting through singular pair interaction potentials of the form $\pm |x-y{|}^{-a}$, which includes the Coulomb and gravitational interactions. We prove that the many-body dynamics of mixed states are well approximated by solutions of the Hartree–Fock and Vlasov equations in terms of Schatten norms. The errors in these approximations are expressed in terms of the expected number of particles, $N$, and the Planck constant, $h$. For cases where $a\in (0,1/2)$, we obtain local-in-time results when $N^{-1/2}\ll h\le N^{-1/3}$. Notably, this leads to the derivation of the Vlasov equation with singular potentials. For cases where $a\in [1/2,1]$, our results hold only within a small time scale or require an $N$-dependent cut-off. A fundamental ingredient in our analysis is the propagation of regularity for solutions to the Hartree–Fock equation uniformly in the Planck constant, which holds for $a\in (0,1]$.