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Scattering for the one dimensional Hartree Fock equation

Cyril Malézé

TL;DR

The paper analyzes the 1D time-dependent Hartree-Fock equation for fermions with small, localized data and a finite-measure interaction $w$, using a random-field reformulation and space-time resonance methods. It reveals a nonlinear cancellation between direct and exchange terms for plane waves, which precludes traditional long-range scattering and guides the asymptotic behavior toward linear dynamics. A fixed-point argument yields global well-posedness in Schatten-1 and scattering to a linear state for the density-matrix formulation, with the random-field profile converging to a limit and a decay rate $t^{-\,\delta}$ in suitable mixed norms. The work further shows that the density-operator dynamics inherit the scattering behavior via a continuity of the covariance mapping, establishing scattering in the density-operator framework as well.

Abstract

We consider the Hartree-Fock equation in 1D, for a small and localised initial data and a finite measure potential. We show that there is no long range scattering due to a nonlinear cancellation between the direct term and the exchange term for plane waves. We employ the framework of space-time resonances that enables us to single out precisely this cancellation and to obtain scattering to linear waves as a consequence.

Scattering for the one dimensional Hartree Fock equation

TL;DR

The paper analyzes the 1D time-dependent Hartree-Fock equation for fermions with small, localized data and a finite-measure interaction , using a random-field reformulation and space-time resonance methods. It reveals a nonlinear cancellation between direct and exchange terms for plane waves, which precludes traditional long-range scattering and guides the asymptotic behavior toward linear dynamics. A fixed-point argument yields global well-posedness in Schatten-1 and scattering to a linear state for the density-matrix formulation, with the random-field profile converging to a limit and a decay rate in suitable mixed norms. The work further shows that the density-operator dynamics inherit the scattering behavior via a continuity of the covariance mapping, establishing scattering in the density-operator framework as well.

Abstract

We consider the Hartree-Fock equation in 1D, for a small and localised initial data and a finite measure potential. We show that there is no long range scattering due to a nonlinear cancellation between the direct term and the exchange term for plane waves. We employ the framework of space-time resonances that enables us to single out precisely this cancellation and to obtain scattering to linear waves as a consequence.
Paper Structure (15 sections, 10 theorems, 132 equations)

This paper contains 15 sections, 10 theorems, 132 equations.

Table of Contents

  1. Introduction
  2. The Hartree-Fock equation and the framework of random fields
  3. Long range scattering for NLS and Hartree-Fock equations
  4. Cancellation for plane waves solutions
  5. Main result
  6. Notations
  7. Organization of the paper
  8. Acknowledgments
  9. Local solution of \ref{['Cauchyprob']}
  10. Proof of Proposition \ref{['prop:bootstrap-estimate']}
  11. Setting the stationary phase argument
  12. Estimate on $R$
  13. Proof of Theorem \ref{['th:principal']}
  14. Proof of scattering for the density operators framework
  15. Sketch of proof of Theorem \ref{['th:local-existence']}

Key Result

Theorem 1.1

Let $w$ be a finite measure and $\gamma_0$ an operator on $L^2(\varmathbb R)$, such that for $\varepsilon>0$ small enough (depending on $w$), there exists a unique global solution $\gamma\in L_t^\infty,\mathfrak S^1$ of Cauchyprob-density-matrices with initial condition $\gamma(t=0)=\gamma_0$, that scatters at infinity, in the sense that there exists $\gamma_\infty\in \mathfrak S^1$ suc

Theorems & Definitions (23)

  • Theorem 1.1
  • Remark 1.1
  • Remark 1.2
  • Theorem 1.2
  • Remark 1.3
  • Remark 1.4
  • Theorem 2.1
  • Remark 2.1
  • Proposition 2.1
  • Lemma 2.1
  • ...and 13 more