On the mean-field and semiclassical limit from quantum N-body dynamics

We study the mean-field and semiclassical limit of the quantum many-body bosonic dynamics with a repulsive δ-type potential $N^{3\beta }V\left(N^{\beta }x\right)$ and a repulsive Coulomb potential on $R^3$, which leads to a macroscopic fluid equation, the Euler-Poisson equation with pressure. We prove quantitative strong convergence of the quantum mass and momentum densities up to the first blow up time of the limiting equation. The proof is based on a modulated energy method, for which a functional inequality is the key ingredient. Different from Golse-Paul [38] in which the sole Coulomb potential case was considered and one could use Serfaty's inequality [64], the δ-type potential's sharp singularity and general profile hinder the application of such inequalities. In this paper, we develop a completely new method, in which Erdős-Schlein-Yau [31], [33], [34] $H^1$-type energy estimate plays an unexpected role, to establish the functional inequality on the δ-type potential for the optimal case $\beta \in (0,1)$.