Asymptotic Stability of Hartree--Fock Homogenous Equilibria in $\mathbb{R}^d$
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arXiv
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| Formato: | Preprint |
| Publicado: |
2026
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| _version_ | 1866914494290067456 |
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| author | Nguyen, Toan T. You, Chanjin |
| author_facet | Nguyen, Toan T. You, Chanjin |
| contents | 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 Schrödinger 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2604_18952 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Asymptotic Stability of Hartree--Fock Homogenous Equilibria in $\mathbb{R}^d$ Nguyen, Toan T. You, Chanjin Analysis of PDEs Mathematical Physics 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 Schrödinger 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. |
| title | Asymptotic Stability of Hartree--Fock Homogenous Equilibria in $\mathbb{R}^d$ |
| topic | Analysis of PDEs Mathematical Physics |
| url | https://arxiv.org/abs/2604.18952 |