{
  "abstract": "We prove an $L\\log L$-endpoint Cwikel estimate on quantum tori $\\mathbb T_θ^d$, i.e., for every $d\\geq 2$ and $x\\in L\\log L(\\mathbb T_θ^d)$, \\begin{equation*} \\big\\|(1-Δ)^{-\\frac{d}{4}}λ(x)(1-Δ)^{-\\frac{d}{4}}\\big\\|_{\\mathcal{L}_{1,\\infty}} \\leq C_d \\Norm{x}_{L\\log L(\\mathbb T_θ^d)}, \\end{equation*} where $λ(x)$ is the left multiplication by $x$. Our argument actually gives an abstract weak Hilbert--Schmidt estimate for multiplication operators in finite von Neumann algebras whose $L_2$-spaces admit orthonormal bases of unitaries. As a consequence, it yields a critical Sobolev embedding $W_2^{d/2}\\hookrightarrow\\exp(L_2)$ on quantum tori. When $θ=0$, our method also gives a simpler proof of the $L\\log L$ Cwikel estimate on classical tori. As an application, we use the $C^*$-algebraic principal symbol map to prove Weyl's laws for sandwich operators \\[ S(1-Δ)^{-d/4}λ(x)(1-Δ)^{-d/4}T, \\] where $x\\in L\\log L(\\mathbb T_θ^d)$ and $S,T\\inΠ$, the $C^*$-algebra generated by $λ(C(\\mathbb T_θ^d))$ and Fourier multipliers with continuous homogeneous symbols of degree zero. We also prove a semiclassical Weyl's law for critical Schrödinger operators with $L\\log L$ potentials.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.20373",
  "author_total": 2,
  "authors": [
    {
      "name": "Siyu Liu"
    },
    {
      "name": "Xinyu Zhang"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/4p23rc0j743xh.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.20373",
  "fulltext_url": null,
  "id": "4p23rc0j743xh",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.20373",
  "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.FA"
    ],
    [
      "arxiv",
      "math.OA"
    ],
    [
      "openalex",
      "2610|Advanced Operator Algebra Research"
    ]
  ],
  "openalex": "W7222226143",
  "pdf_url": "https://arxiv.org/pdf/2610.20373",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-20",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/4p23rc0j743xh.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:34:22Z",
      "native_id": "2610.20373",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.20373"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.FA",
      "subfield": "analysis-pdes"
    }
  ],
  "title": "Cwikel estimates and Weyl's laws for $L\\log L$ potentials on quantum tori",
  "title_text": "Cwikel estimates and Weyl's laws for Llog L potentials on quantum tori",
  "type": "preprint",
  "updated_at": "2026-10-30T12:34:22Z",
  "url": "https://research.draconitic.com/paper/4p23rc0j743xh",
  "venue": "arXiv",
  "versions": []
}