Global Semiclassical Limit from Hartree to Vlasov Equation for Concentrated Initial Data

Abstract

We prove a quantitative and global in time semiclassical limit from the Hartree to the Vlasov equation in the case of a singular interaction potential in dimension $d\ge 3$, including the case of a Coulomb singularity in dimension $d=3$. This result holds for initial data concentrated enough in the sense that some space moments are initially sufficiently small. As an intermediate result, we also obtain quantitative semiclassical bounds on the space and velocity moments of even order and the asymptotic behavior of the spatial density due to dispersion effects.

26 pages. V3: Corrections and precisions in the proofs

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• On the $L^2$ Rate of Convergence in the Limit from the Hartree to the Vlasov$\unicode{x2013}$Poisson Equation
• Propagation of moments for large data and semiclassical limit to the relativistic Vlasov equation
• Uniqueness criteria for the Vlasov--Poisson system and applications to semiclassical analysis
• Effective Dynamics of Interacting Fermions from Semiclassical Theory to the Random Phase Approximation
• Enhanced Stability in Quantum Optimal Transport Pseudometrics: From Hartree to Vlasov-Poisson