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Scattering for the positive density Hartree equation

Antoine Borie, Sonae Hadama, Julien Sabin

TL;DR

This work proves the asymptotic stability and linear scattering around homogeneous stationary states for the positive density Hartree equation in $\mathbb{R}^d$, $d\ge3$. The authors develop a robust framework that combines density-matrix fractional Leibniz rules with Christ-Kiselev lemmas in Schatten spaces to obtain optimal Sobolev/ Schatten exponents for small perturbations, even with rough potentials including $\delta$-type interactions. The analysis hinges on a fixed-point scheme for the density perturbation $\rho_Q$ and a Penrose-stability condition ensuring invertibility of the linearized response, together with generalized Strichartz bounds for the evolving propagator $U_V$. The result yields global existence and linear scattering around $g(-i\nabla)$ for initial data in the sharp space $\mathcal{H}^{d/2-1,2d/(d+1)}$, with scattering described by $e^{it\Delta}Q_\pm e^{-it\Delta}$ in the corresponding Schatten topology, advancing prior work to $d\ge3$ and weaker interaction assumptions. Overall, the paper provides a cohesive, technical toolkit—centered on fractional Leibniz rules and CK lemmas in Schatten spaces—that could extend to related mean-field/density-matrix problems and Vlasov-Hartree-type dynamics.

Abstract

We study the asymptotic stability for large times of homogeneous stationary states for the nonlinear Hartree equation for density matrices in Rd for d\geq3. We can reach both the optimal Sobolev and Schatten exponents for the initial data, with a wide class of interaction potentials w (under the sole assumption that w is bounded, including in particular delta potentials). Our method relies on fractional Leibniz rules for density matrices to deal with the fractional critical Sobolev regularity s = d/2 -1 for odd d, as well as Christ-Kiselev lemmas in Schatten spaces.

Scattering for the positive density Hartree equation

TL;DR

This work proves the asymptotic stability and linear scattering around homogeneous stationary states for the positive density Hartree equation in , . The authors develop a robust framework that combines density-matrix fractional Leibniz rules with Christ-Kiselev lemmas in Schatten spaces to obtain optimal Sobolev/ Schatten exponents for small perturbations, even with rough potentials including -type interactions. The analysis hinges on a fixed-point scheme for the density perturbation and a Penrose-stability condition ensuring invertibility of the linearized response, together with generalized Strichartz bounds for the evolving propagator . The result yields global existence and linear scattering around for initial data in the sharp space , with scattering described by in the corresponding Schatten topology, advancing prior work to and weaker interaction assumptions. Overall, the paper provides a cohesive, technical toolkit—centered on fractional Leibniz rules and CK lemmas in Schatten spaces—that could extend to related mean-field/density-matrix problems and Vlasov-Hartree-type dynamics.

Abstract

We study the asymptotic stability for large times of homogeneous stationary states for the nonlinear Hartree equation for density matrices in Rd for d\geq3. We can reach both the optimal Sobolev and Schatten exponents for the initial data, with a wide class of interaction potentials w (under the sole assumption that w is bounded, including in particular delta potentials). Our method relies on fractional Leibniz rules for density matrices to deal with the fractional critical Sobolev regularity s = d/2 -1 for odd d, as well as Christ-Kiselev lemmas in Schatten spaces.
Paper Structure (29 sections, 36 theorems, 484 equations)

This paper contains 29 sections, 36 theorems, 484 equations.

Table of Contents

  1. Context
  2. Main results
  3. Comparison with existing results
  4. Proof strategy and novelties
  5. Structure of the article
  6. Preliminaries
  7. Notations
  8. Fixed point setting
  9. Linear stability
  10. Strichartz estimates for the free Laplacian
  11. Generalized Strichartz estimates
  12. Fractional Leibniz rules
  13. Notations and key results
  14. Fractional Leibniz decompositions
  15. Part 1:
  16. ...and 14 more sections

Key Result

Theorem 1

Let $d\geqslant3$. Let $w\in\mathcal{S}'({ {\mathbb R} }^d)$ be such that $\widehat{w}$ is even, continuous and bounded. Let $g\in L^1({ {\mathbb R} }^d,{ {\mathbb R} }_+)$ be such that $\langle\xi\rangle^{2(d-2)}g(\xi)\in L^{ {\infty}}_\xi({ {\mathbb R} }^d)$ and $\sup_{\omega\in\mathbb{S}^{d-1}}\i

Theorems & Definitions (92)

  • Theorem 1
  • Remark 1
  • Remark 2
  • Remark 3
  • Definition 1.1: Quantum Penrose stability condition
  • Remark 1.2
  • Remark 1.3
  • Proposition 1.4
  • proof
  • Remark 1.5
  • ...and 82 more