Convergence towards the Vlasov-Poisson Equation from the $N$-Fermionic Schr\"odinger Equation

We consider the quantum dynamics of $N$ interacting fermions in the large $N$ limit. The particles in the system interact with each other via repulsive interaction that is regularized Coulomb with a polynomial cutoff with respect to $N$. From the quantum system, we derive the Vlasov-Poisson system by simultaneously estimating the semiclassical and mean-field residues in terms of the Husimi measure.