{
  "abstract": "We study the time-dependent Hartree--Fock equation in $\\R^3$ with physical Coulomb interactions in both direct and exchange channels, with exchange coupling $η$, linearized about homogeneous equilibria $γ_f=g(-i\\nabla)$ having compact momentum support. Nguyen and You showed that, without exchange, two purely imaginary plasmon poles $\\pm iτ_0(|k|)$ exist below a survival threshold $κ_0>0$. Existing nonlinear Hartree--Fock theory requires a short-range direct interaction and a smooth small exchange kernel, excluding the present Coulomb setting. On every compact band $0<k_-\\le|k|\\le k_+<κ_0$, we prove unconditionally for Coulomb exchange that, for small $|η|$, the poles persist, remain simple and purely imaginary, and depend real-analytically on $η$. Their first-order shift $τ_X$ splits into explicit static self-energy and dynamic vertex corrections. Each pole yields an exact undamped mode with a closed-form unit-density eigenfunction, embedded in the fiber generator's essential spectrum. The causal density Green function splits into an explicit undamped sine wave and a remainder whose Laplace transform is holomorphic near the poles, while the plasmon channel satisfies uniform $t^{-3/2}$ dispersive decay and endpoint Strichartz estimates. A first-order Ward-type cancellation gives $|τ_X(r)|\\le Cr^2$ down to $r=0$; hence exchange leaves the plasma gap unchanged to first order, and we obtain a closed formula for the stiffness correction. Numerics for a $C^9$ smoothed Fermi ball support the analysis: self-energy and vertex terms cancel within $1.2\\%$ on the computed range, and exchange softens the dispersion most strongly near $0.7\\,κ_0$.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2608.29671",
  "author_total": 1,
  "authors": [
    {
      "name": "Yuya Dan"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/f06hf0qj0519j.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2608.29671",
  "fulltext_url": null,
  "id": "f06hf0qj0519j",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2608.29671",
  "language": null,
  "licence": "CC-BY-4.0",
  "licence_label": "CC BY 4.0",
  "licence_source": "article",
  "licence_url": "http://creativecommons.org/licenses/by/4.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "math.AP"
    ],
    [
      "openalex",
      "2610|Advanced Mathematical Physics Problems"
    ]
  ],
  "openalex": "W7204904871",
  "pdf_url": "https://arxiv.org/pdf/2608.29671",
  "pmcid": null,
  "pmid": null,
  "published": "2026-08-30",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/f06hf0qj0519j.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:17:48Z",
      "native_id": "2608.29671",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2608.29671"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.AP",
      "subfield": "analysis-pdes"
    }
  ],
  "title": "Plasmon poles for the linearized Coulomb--Hartree--Fock equation",
  "title_text": "Plasmon poles for the linearized Coulomb--Hartree--Fock equation",
  "type": "preprint",
  "updated_at": "2026-10-30T12:17:48Z",
  "url": "https://research.draconitic.com/paper/f06hf0qj0519j",
  "venue": "arXiv",
  "versions": []
}