Convergence rate towards the fractional Hartree equation with singular potentials in higher Sobolev trace norms

In this work, we show convergence of the N-particle bosonic Schrödinger equation towards the Hartree equation. Hereby, we extend the results of [I. Anapolitanos and M. Hott, A simple proof of convergence to the Hartree dynamics in Sobolev trace norms, J. Math. Phys. 57(12) (2016) 122108; I. Anapolitanos, M. Hott and D. Hundertmark, Derivation of the Hartree equation for compound Bose gases in the mean field limit, Rev. Math. Phys. 29(07) (2017) 1750022]. We first consider the semi-relativistic Hartree equation in the defocusing and the focusing cases. We show that Pickl's projection method [P. Pickl, Derivation of the time dependent Gross-Pitaevskii equation without positivity condition on the interaction, J. Statist. Phys. 140(1) (2010) 76-89; P. Pickl, A simple derivation of mean field limits for quantum systems, Lett. Math. Phys. 97(2) (2011) 151-164; P. Pickl, Derivation of the time dependent Gross-Pitaevskii equation with external fields, Rev. Math. Phys. 27(1) (2015) 1550003], can be adapted to this problem. Next, we extend this result to the case of fractional Hartree equations with potentials that are more singular than the Coulomb potential. Finally, in the non-relativistic case, we derive the Hartree equation assuming only L2 initial data for potentials with a quantitative bound on the convergence rate.