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In this paper we establish almost-optimal stability estimates in quantumoptimal transport pseudometrics for the semiclassical limit of the Hartreedynamics to the Vlasov-Poisson equation, in the regime where the solutions havebounded densities. We combine Golse and Paul's method from [Arch. Ration. Mech.Anal. 223:57-94, 2017], which uses a semiclassical version of the optimaltransport distance and which was adapted to the case of the Coulomb andgravitational interactions by the second author in [J. Stat. Phys. 177:20-60,2019], with a new approach developed by the first author in [Arch. Ration.Mech. Anal. 244:27-50, 2022] to quantitatively improve stability estimates inkinetic theory.

Enhanced Stability in Quantum Optimal Transport Pseudometrics: From Hartree to Vlasov-Poisson