Scattering for the positive density Hartree equation

We study the asymptotic stability for large times of homogeneous stationary states for the nonlinear Hartree equation for density matrices in Rd for d\geq3. We can reach both the optimal Sobolev and Schatten exponents for the initial data, with a wide class of interaction potentials w (under the sole assumption that w is bounded, including in particular delta potentials). Our method relies on fractional Leibniz rules for density matrices to deal with the fractional critical Sobolev regularity s = d/2 -1 for odd d, as well as Christ-Kiselev lemmas in Schatten spaces.