{
  "abstract": "We prove the refined Koornwinder--Kostenko--Teschl conjecture. For the normalized weighted Jacobi function and all $n\\in\\mathbb N\\_0$, $α,β\\ge0$, and $-1\\le x\\le1$, $$ |g\\_n^{(α,β)}(x)| \\le \\left[ \\frac{(n+1)(n+α+β+1)} {(n+α+1)(n+β+1)} \\right]^{1/4}\\le1. $$ The proof combines a central contour estimate with Sturm--Sonin localization and an exact inverse moment. It also yields an $n$-uniform extreme-lobe theorem and a sharp canonical-product first-lobe principle. Applied to the discrete Laguerre evolution, the estimate gives the optimal positive-parameter decay. For $-1<α\\le0$, a complementary one-sided Jacobi inequality gives the exact norm $\\|\\mathrm{e}^{-\\mathrm{i} tH_α}\\|_{\\ell^1\\to\\ell^\\infty} =(1+t^2)^{-(1+α)/2}$. Thus the large-time decay exponent is $\\min\\{1,1+α\\}$ for every $α>-1$. Bessel, Laguerre, and Darboux scaling limits show that the temporal and fixed-diagonal exponents are optimal.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2608.30486",
  "author_total": 1,
  "authors": [
    {
      "name": "Yu-Tian Li"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/68zrxjk8hyfv4.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2608.30486",
  "fulltext_url": null,
  "id": "68zrxjk8hyfv4",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2608.30486",
  "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.CA"
    ],
    [
      "arxiv",
      "math.SP"
    ],
    [
      "openalex",
      "2604|Mathematical functions and polynomials"
    ]
  ],
  "openalex": "W7204917604",
  "pdf_url": "https://arxiv.org/pdf/2608.30486",
  "pmcid": null,
  "pmid": null,
  "published": "2026-08-31",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/68zrxjk8hyfv4.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:17:48Z",
      "native_id": "2608.30486",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2608.30486"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.CA",
      "subfield": "analysis-pdes"
    }
  ],
  "title": "The Koornwinder--Kostenko--Teschl Conjecture for Jacobi Polynomials and the Discrete Laguerre Phase Transition",
  "title_text": "The Koornwinder--Kostenko--Teschl Conjecture for Jacobi Polynomials and the Discrete Laguerre Phase Transition",
  "type": "preprint",
  "updated_at": "2026-10-30T12:17:48Z",
  "url": "https://research.draconitic.com/paper/68zrxjk8hyfv4",
  "venue": "arXiv",
  "versions": []
}