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From many-body quantum dynamics to the Hartree-Fock and Vlasov equations with singular potentials

Jacky J. Chong, Laurent Lafleche, Chiara Saffirio

Abstract

We obtain the combined mean-field and semiclassical limit from the $N$-body Schrödinger equation for fermions interacting via singular potentials. To obtain the result, we first prove the uniformity in Planck's constant $h$ propagation of regularity for solutions to the Hartree$\unicode{x2013}$Fock equation with singular pair interaction potentials of the form $\pm |x-y|^{-a}$, including the Coulomb and gravitational interactions. In the context of mixed states, we use these regularity properties to obtain quantitative estimates on the distance between solutions to the Schrödinger equation and solutions to the Hartree$\unicode{x2013}$Fock and Vlasov equations in Schatten norms. For $a\in(0,1/2)$, we obtain local-in-time results when $N^{-1/2} \ll h \leq N^{-1/3}$. In particular, it leads to the derivation of the Vlasov equation with singular potentials. For $a\in[1/2,1]$, our results hold only on a small time scale, or with an $N$-dependent cutoff.

From many-body quantum dynamics to the Hartree-Fock and Vlasov equations with singular potentials

Abstract

We obtain the combined mean-field and semiclassical limit from the -body Schrödinger equation for fermions interacting via singular potentials. To obtain the result, we first prove the uniformity in Planck's constant propagation of regularity for solutions to the HartreeFock equation with singular pair interaction potentials of the form , including the Coulomb and gravitational interactions. In the context of mixed states, we use these regularity properties to obtain quantitative estimates on the distance between solutions to the Schrödinger equation and solutions to the HartreeFock and Vlasov equations in Schatten norms. For , we obtain local-in-time results when . In particular, it leads to the derivation of the Vlasov equation with singular potentials. For , our results hold only on a small time scale, or with an -dependent cutoff.

Paper Structure

This paper contains 47 sections, 49 theorems, 537 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Background
  3. The Equations
  4. Mean-field and Semiclassical Scalings
  5. State of the Art
  6. The Classical Mean-Field Limit
  7. Combined Mean-Field and Semiclassical Limits
  8. Quantum Mean-Field Limit
  9. Semiclassical Limit
  10. Constraints on the Scalings
  11. Functional Spaces
  12. Semiclassical Spaces
  13. Fermionic Fock Space
  14. Main Results
  15. Propagation of Regularity
  16. ...and 32 more sections

Key Result

Theorem 3.1

Let $a\in (0,1]$, $m_n=1+\left\lvert \boldsymbol{p} \right\rvert^n$ with $n\in 2\mathbb N$ satisfying $n\geq 6$ and $\boldsymbol{\rho}$ be a solution to the Hartree--Fock equation eq:Hartree--Fock with initial condition $\boldsymbol{\rho}^\mathrm{in}\in \mathcal{L}^\infty(m_n)$ satisfying eq:scaling Then there exists $T>0$ such that uniformly in $\hbar \in (0, 1)$.

Figures (1)

  • Figure 1: The different scalings for the combined mean-field and semiclassical limits. The dashed (red) curve corresponds to the equation $h = N^{-1/2}$ and the continuous (blue) curve to $h = N^{-1/3}$.

Theorems & Definitions (110)

  • Theorem 3.1: Propagation of regularity
  • Remark 3.1
  • Remark 3.2: On the initial data of the Hartree equation
  • Theorem 3.2: Mean-field limit
  • Remark 3.3
  • Remark 3.4
  • Theorem 3.3: Combined mean-field and semiclassical limits
  • Remark 3.5
  • Remark 3.6
  • Theorem 4.1
  • ...and 100 more