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arXiv · 2604.18952

Asymptotic Stability of Hartree--Fock Homogenous Equilibria in $\mathbb{R}^d$

Toan T. Nguyen·Chanjin You
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Abstract

In this paper, we establish nonlinear Landau damping and asymptotic stability of a large class of translation-invariant steady solutions to the time-dependent Hartree--Fock equations in the presence of an {\em off-diagonal exchange operator}, which arises naturally in the meanfield theory of a large fermionic system, in the whole space $\mathbb{R}^d$, $d\ge 3$. Despite being a sub-order operator, the inclusion of the exchange term disturbs the classical Schr\"odinger dispersion and causes a complex linear response from the background electrons to the space density whose dispersion relation is no longer a Fourier multiplier as in the classical Vlasov and Hartree theory. In addition, the group velocity of each elementary waves involves a mixture of all other Fourier modes, leading to delicate {\em momentum-dependent echo resonances}. To overcome the issues, we develop a nonlinear iterative scheme that relies on a detailed resolvent analysis, makes use of a transport type dispersion in Fourier spaces, and propagates phase mixing and Landau damping in weighted $L^\infty_{k,p}$ norms.

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Toan T. Nguyen, Chanjin You. 2026-04-21. Asymptotic Stability of Hartree--Fock Homogenous Equilibria in $\mathbb{R}^d$. https://arxiv.org/abs/2604.18952

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