On the derivation of the Hartree equation from the N-body Schrödinger equation: Uniformity in the Planck constant

In this paper the Hartree equation is derived from the N-body Schrödinger equation in the mean-field limit, with convergence rate estimates that are uniform in the Planck constant ħ. Specifically, we consider the two following cases: (a) Töplitz initial data and Lipschitz interaction forces, and (b) analytic initial data and interaction potential, over short time intervals independent of ħ. The convergence rates in these two cases are 1/log⁡log⁡N and 1/N respectively. The treatment of the second case is entirely self-contained and all the constants appearing in the final estimate are explicit. It provides a derivation of the Vlasov equation from the N-body classical dynamics using BBGKY hierarchies instead of empirical measures.