Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Stability of homogeneous equilibria of the Hartree-Fock equation, for its equivalent formulation for random fields

Charles Collot, Elena Danesi, Anne-Sophie de Suzzoni, Cyril Malézé

TL;DR

This paper analyzes the time-dependent Hartree-Fock equation in a random-field formulation and proves nonlinear asymptotic stability and scattering of homogeneous equilibria in high dimensions ($d\ge 4$), explicitly including the exchange term. The authors reformulate the dynamics around a nonlocalised equilibrium $Y_f$ as a coupled system for the perturbation $Z$ and the two-point correlation $V$, and solve it via a Banach fixed-point argument in carefully chosen function spaces that encode dispersive and smoothing properties. Central to the approach are Strichartz estimates for the linearized flow around the equilibrium, explicit representation formulas for the linear terms, and bilinear estimates that control the quadratic nonlinearities, including the exchange contributions. Under small perturbations of the equilibrium and mild regularity/ellipticity assumptions on the phase $\theta$, they obtain global existence, uniqueness in a perturbative regime, and scattering to a linear state, thereby extending stability results to the full Hartree-Fock dynamics and bridging mean-field theory with random-field formulations. The results have potential implications for mean-field limits of large fermionic systems and for density-functional-type models that retain exchange effects beyond the reduced Hartree approximation.

Abstract

The Hartree-Fock equation admits homogeneous states that model infinitely many particles at equilibrium. We prove their asymptotic stability in large dimensions, under assumptions on the linearised operator. Perturbations are moreover showed to scatter to linear waves. We obtain this result for the equivalent formulation of the Hartree-Fock equation in the framework of random fields. The main novelty is to study the full Hartree-Fock equation, including for the first time the exchange term in the study of these stationary solutions.

Stability of homogeneous equilibria of the Hartree-Fock equation, for its equivalent formulation for random fields

TL;DR

This paper analyzes the time-dependent Hartree-Fock equation in a random-field formulation and proves nonlinear asymptotic stability and scattering of homogeneous equilibria in high dimensions (), explicitly including the exchange term. The authors reformulate the dynamics around a nonlocalised equilibrium as a coupled system for the perturbation and the two-point correlation , and solve it via a Banach fixed-point argument in carefully chosen function spaces that encode dispersive and smoothing properties. Central to the approach are Strichartz estimates for the linearized flow around the equilibrium, explicit representation formulas for the linear terms, and bilinear estimates that control the quadratic nonlinearities, including the exchange contributions. Under small perturbations of the equilibrium and mild regularity/ellipticity assumptions on the phase , they obtain global existence, uniqueness in a perturbative regime, and scattering to a linear state, thereby extending stability results to the full Hartree-Fock dynamics and bridging mean-field theory with random-field formulations. The results have potential implications for mean-field limits of large fermionic systems and for density-functional-type models that retain exchange effects beyond the reduced Hartree approximation.

Abstract

The Hartree-Fock equation admits homogeneous states that model infinitely many particles at equilibrium. We prove their asymptotic stability in large dimensions, under assumptions on the linearised operator. Perturbations are moreover showed to scatter to linear waves. We obtain this result for the equivalent formulation of the Hartree-Fock equation in the framework of random fields. The main novelty is to study the full Hartree-Fock equation, including for the first time the exchange term in the study of these stationary solutions.
Paper Structure (20 sections, 18 theorems, 270 equations)

This paper contains 20 sections, 18 theorems, 270 equations.

Table of Contents

  1. Introduction
  2. The Hartree-Fock equation in the framework of random fields
  3. Stability of non-localised equilibria for the reduced Hartree-Fock equation
  4. Main result
  5. Formal derivation of the equation from the N body problem
  6. Equilibria
  7. Acknowledgments
  8. Notation
  9. Setting the contraction argument
  10. Linear estimates
  11. Strichartz estimates
  12. Representation formulas for the linear terms
  13. The issue of low frequency regularity in the linear response
  14. Estimates and invertibility on the last linear terms
  15. Bilinear estimates
  16. ...and 5 more sections

Key Result

Theorem 1.1

Let $d\geq 4$. Let $w$ be an even Borel measure with $\langle y \rangle w$ a finite measure, and $g=f^2$ that satisfies $g \in W^{3,1}\cap W^{3,\infty}$ and $\langle \xi\rangle^{2\lceil s_c \rceil}g\in W^{2,1}$, be such that $\theta$ given by id:def-theta satisfies id:hp-regularite-tilde-theta and i the following holds true. There exists $\delta>0$ such that if the Cauchy problem Cauchyprob with

Theorems & Definitions (38)

  • Theorem 1.1
  • Remark 1.1
  • Remark 3.1
  • Proposition 4.1: Strichartz estimates
  • Remark 4.1
  • proof
  • Theorem 4.1
  • Theorem 4.2
  • Remark 4.2
  • Corollary 4.1
  • ...and 28 more