{
  "abstract": "The finite-rank fractional Lieb--Thirring inequality provides an estimate on the Riesz sum of the first $N$ negative eigenvalues of the fractional Schrödinger operator $(-Δ)^s-V$ in terms of the $L^{γ+\\frac{d}{2s}}(\\mathbb R^d)$ norm of the potential $V$, where the Riesz exponent $γ>\\max\\left\\{0,1-d/(2s)\\right\\}$. We prove the existence of optimizers for every finite rank by establishing a result on the spectrum in the case of dichotomy. We further characterize the spectral gap and qualitative properties of optimizers, including the Euler--Lagrange equation, regularity, and asymptotic behavior at infinity. In particular, under the condition $γ+{d}/{(2s)}\\geq 3+{4s}/{d}$, we show that all normalized maximizing sequences are precompact modulo translations. Moreover, we derive a strict binding inequality for the optimal constants in the finite-rank setting, which implies that the optimal fractional Lieb--Thirring constant, if it exists, cannot be attained by a potential with finitely many negative eigenvalues.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.21716",
  "author_total": 2,
  "authors": [
    {
      "name": "Lu Chen"
    },
    {
      "name": "Lianfeng Yang"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/cc18mxtyjptrp.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.21716",
  "fulltext_url": null,
  "id": "cc18mxtyjptrp",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.21716",
  "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",
      "2610|Spectral Theory in Mathematical Physics"
    ]
  ],
  "openalex": "W7222754778",
  "pdf_url": "https://arxiv.org/pdf/2610.21716",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-15",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/cc18mxtyjptrp.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:29:54Z",
      "native_id": "2610.21716",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.21716"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.AP",
      "subfield": "analysis-pdes"
    }
  ],
  "title": "Optimizers for the finite-rank fractional Lieb--Thirring inequality",
  "title_text": "Optimizers for the finite-rank fractional Lieb--Thirring inequality",
  "type": "preprint",
  "updated_at": "2026-10-30T12:29:54Z",
  "url": "https://research.draconitic.com/paper/cc18mxtyjptrp",
  "venue": "arXiv",
  "versions": []
}