Global-in-time Semiclassical Regularity for the Hartree-Fock Equation

Abstract

For arbitrarily large times $T>0$, we prove the uniform-in-$\hslash$ propagation of semiclassical regularity for the solutions to the Hartree$\unicode{x2013}$Fock equation with singular interactions of the form $V(x)=\pm |x{|}^{-a}$ where $a\in (0,\frac{1}{2})$. As a byproduct of this result, we extend to arbitrarily long times the derivation of the Hartree$\unicode{x2013}$Fock and the Vlasov equations from the many-body dynamics provided in [J. Chong, L. Lafleche, C. Saffirio: arXiv:2103.10946 (2021)].

11 pages

References in corpus (3)

• Mean field evolution of fermions with Coulomb interaction
• Semiclassical limit to the Vlasov equation with inverse power law potentials
• From many-body quantum dynamics to the Hartree-Fock and Vlasov equations with singular potentials

Cited by in corpus (4)

• On Quantum Sobolev Inequalities
• On the $L^2$ Rate of Convergence in the Limit from the Hartree to the Vlasov$\unicode{x2013}$Poisson Equation
• Enhanced Stability in Quantum Optimal Transport Pseudometrics: From Hartree to Vlasov-Poisson
• Stability of homogeneous equilibria of the Hartree-Fock equation, for its equivalent formulation for random fields