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

Jacky J. Chong, Laurent Lafleche, Chiara Saffirio

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)].

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

Abstract

For arbitrarily large times , we prove the uniform-in- propagation of semiclassical regularity for the solutions to the HartreeFock equation with singular interactions of the form where . As a byproduct of this result, we extend to arbitrarily long times the derivation of the HartreeFock and the Vlasov equations from the many-body dynamics provided in [J. Chong, L. Lafleche, C. Saffirio: arXiv:2103.10946 (2021)].
Paper Structure (5 sections, 9 theorems, 66 equations)

This paper contains 5 sections, 9 theorems, 66 equations.

Table of Contents

  1. Introduction and Main Result
  2. Propagation of Moments
  3. The Hartree Equation
  4. The Hartree--Fock Equation
  5. Proof of Theorem \ref{['thm:prop-reg-grad']}

Key Result

Theorem 1.1

Let $a\in$0,12$$, $n\in 2\mathbb N$ be an even integer and $\boldsymbol{\rho}$ be a solution to the Hartree--Fock equation eq:HF with initial datum $\boldsymbol{\rho}^\mathrm{in}\in\mathcal{L}^\infty(\mathsf m_n)$ satisfying eq:fermionic-bounds and such that for $q\in [2, \infty)$, and with moments of order strictly larger than $\frac{3}{1-a}$n+a+1$$ bounded uniformly in $\hbar$. Then uniformly

Theorems & Definitions (18)

  • Theorem 1.1
  • Theorem 2.1
  • Remark 2.1
  • Remark 2.2
  • Lemma 2.1
  • proof
  • Lemma 2.2
  • Corollary 2.1
  • Remark 2.3
  • proof : Proof of Theorem \ref{['thm:propag_moments_HF']}
  • ...and 8 more