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

Abstract In this paper the Hartree equation is derived from the N-body Schrodinger 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) Toplitz 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.

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