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Weyl laws for interacting particles

Ngoc Nhi Nguyen

TL;DR

The paper develops Weyl laws for interacting fermions in the grand-canonical Hartree–Fock framework under confining potentials, establishing integrated and pointwise first-order terms in the semiclassical limit $\hbar\to0$. It connects Hartree–Fock minimizers to Thomas–Fermi densities through a variational chain involving Vlasov and TF functionals, and extends prior canonical results to the grand-canonical setting, using Conlon’s approach for pointwise limits. The key contributions include the weak and pointwise semiclassical limits of HF densities, the convergence of HF ground-state energy to the TF energy, and a rigorous link between quantum many-body states and semiclassical TF descriptions in the presence of repulsive interactions. The work provides a rigorous bridge between quantum mean-field descriptions and classical-like density theories, with implications for understanding interacting fermions in traps and for deriving accurate semiclassical approximations in many-body quantum systems.

Abstract

We study grand-canonical interacting fermionic systems in the mean-field regime, in a trapping potential. We provide the first order term of integrated and pointwise Weyl laws, but in the case with interaction. More precisely, we prove the convergence of the densities of the grand-canonical Hartree-Fock ground state to the Thomas-Fermi ground state in the semiclassical limit $\hbar\to 0$. For the proof, we write the grand-canonical version of the results of Fournais, Lewin and Solovej (Calc. Var. Partial Differ. Equ., 2018) and of Conlon (Commun. Math. Phys., 1983).

Weyl laws for interacting particles

TL;DR

The paper develops Weyl laws for interacting fermions in the grand-canonical Hartree–Fock framework under confining potentials, establishing integrated and pointwise first-order terms in the semiclassical limit . It connects Hartree–Fock minimizers to Thomas–Fermi densities through a variational chain involving Vlasov and TF functionals, and extends prior canonical results to the grand-canonical setting, using Conlon’s approach for pointwise limits. The key contributions include the weak and pointwise semiclassical limits of HF densities, the convergence of HF ground-state energy to the TF energy, and a rigorous link between quantum many-body states and semiclassical TF descriptions in the presence of repulsive interactions. The work provides a rigorous bridge between quantum mean-field descriptions and classical-like density theories, with implications for understanding interacting fermions in traps and for deriving accurate semiclassical approximations in many-body quantum systems.

Abstract

We study grand-canonical interacting fermionic systems in the mean-field regime, in a trapping potential. We provide the first order term of integrated and pointwise Weyl laws, but in the case with interaction. More precisely, we prove the convergence of the densities of the grand-canonical Hartree-Fock ground state to the Thomas-Fermi ground state in the semiclassical limit . For the proof, we write the grand-canonical version of the results of Fournais, Lewin and Solovej (Calc. Var. Partial Differ. Equ., 2018) and of Conlon (Commun. Math. Phys., 1983).
Paper Structure (35 sections, 29 theorems, 227 equations)

This paper contains 35 sections, 29 theorems, 227 equations.

Table of Contents

  1. Introduction
  2. Notation
  3. Hartree-Fock setting
  4. Thomas-Fermi and Vlasov settings
  5. Main results
  6. Whole many-body grand-canonical setting
  7. Useful preliminary properties
  8. About the Hartree-Fock functional
  9. Setting and well-posedness of the Hartree-Fock energy
  10. Coercivity of the Hartree-Fock energy
  11. Properties on almost-minimizers and link with reduced-Hartree-Fock model
  12. Existence of Hartree-Fock minimizers
  13. Nonlinear equation of the minimizers
  14. $\triangleright$ Step 1.
  15. $\triangleright$ Step 2.
  16. ...and 20 more sections

Key Result

Theorem 1

Let $d\ge1$. Under suitable assumptions on $V$ and $w$ (in particular, assuming that $w$ is repulsive), any minimizer $\gamma_\hbar$ of $\mathcal{E}_{\hbar,V-E,w}^{\rm HF}$ satisfies (up to a subsequence) the integrated Weyl law as well as the pointwise Weyl law for all $x\in\mathbb{R}^d$, where $\rho_{\rm TF}$ is a minimizer of $\mathcal{E}^{\rm TF}_{V-E,w}$.

Theorems & Definitions (55)

  • Theorem 1
  • Remark 1.3
  • Remark 1.5
  • Definition 1.6
  • Theorem 2: Weak semiclassical limit of the density
  • Remark 1.7: Assumptions on $w$
  • Theorem 3: Integrated Weyl law
  • Theorem 4
  • Remark 1.8
  • Remark 1.9
  • ...and 45 more