{
  "abstract": "We study optimal potentials for the finite-rank fractional Lieb--Thirring problem at the eigenvalue-counting endpoint, for \\(0<s<1\\) and \\(d>2s\\). Using the zero-energy Birman--Schwinger formulation, we prove a profile decomposition of normalized optimizing sequences with exact splitting of rank and potential mass, and strong approximation in \\(L^{d/(2s)}\\). Strict subadditivity yields existence and compactness modulo translations and dilations. At rank two, the optimal constant equals the rank-one constant, the supremum is not attained, and every normalized optimizing sequence splits into two Sobolev potentials. We also prove strict increase of the minimal potential masses and a uniform spectral gap for optimizing sequences. For each optimizer, we derive an Euler--Lagrange system and establish boundedness, local Hölder regularity and \\(V(x)=|x|^{-4s}(c_\\infty+o(1))\\) as \\(|x|\\to\\infty\\), with \\(c_\\infty\\geq0\\). The conformal orbit of every optimizer contains a representative with a strictly positive asymptotic coefficient.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.41148",
  "author_total": 2,
  "authors": [
    {
      "name": "Zhuocheng Li"
    },
    {
      "name": "Xiaoyu Zeng"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/0efskba6qcx94.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.41148",
  "fulltext_url": null,
  "id": "0efskba6qcx94",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.41148",
  "language": null,
  "licence": "LicenseRef-arXiv-nonexclusive-1.0",
  "licence_label": "arXiv licence",
  "licence_source": "article",
  "licence_url": "http://arxiv.org/licenses/nonexclusive-distrib/1.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "math.AP"
    ]
  ],
  "openalex": null,
  "pdf_url": "https://arxiv.org/pdf/2610.41148",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-30",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/0efskba6qcx94.ris",
  "sources": [
    {
      "first_seen": "2026-11-02T12:21:20Z",
      "native_id": "2610.41148",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.41148"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.AP",
      "subfield": "analysis-pdes"
    }
  ],
  "title": "Critical Optimizers for the Finite-Rank Fractional Lieb--Thirring Inequality",
  "title_text": "Critical Optimizers for the Finite-Rank Fractional Lieb--Thirring Inequality",
  "type": "preprint",
  "updated_at": "2026-11-04T15:20:52Z",
  "url": "https://research.draconitic.com/paper/0efskba6qcx94",
  "venue": "arXiv",
  "versions": []
}