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

TL;DR

This paper establishes uniform semiclassical regularity for solutions to the Hartree-Fock equation with singular interactions over arbitrary long times, extending the derivation of related mean-field equations from many-body quantum dynamics.

Contribution

It proves global-in-time semiclassical regularity for the Hartree-Fock equation with singular potentials, extending previous derivations to arbitrary long times.

Findings

Uniform-in-$$ regularity propagation for solutions.

Extension of derivation of Hartree-Fock and Vlasov equations to long times.

Handling of singular interactions with $a \,<\,\frac{1}{2}$.

Abstract

For arbitrarily large times $T>0$, we prove the uniform-in-$\hbar$ 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 $a\in(0,\frac12)$. 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)].