The Koornwinder--Kostenko--Teschl Conjecture for Jacobi Polynomials and the Discrete Laguerre Phase Transition

Abstract

We prove the refined Koornwinder--Kostenko--Teschl conjecture. For the normalized weighted Jacobi function and all n∈ℕ_0, α,β≥0, and −1≤ x≤1, |g_n^(α,β)(x)| ≤ [ ((n+1)(n+α+β+1))/((n+α+1)(n+β+1)) ]^1/4≤1. 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≤0, a complementary one-sided Jacobi inequality gives the exact norm |e^{−i tH_α}|_{ℓ¹→ℓ^∞} =(1+t²)^−(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.

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BibTeX RIS

BibTeX

@misc{li2026koornwinderkostenkoteschl,
  title = {{The Koornwinder--Kostenko--Teschl Conjecture for Jacobi Polynomials and the Discrete Laguerre Phase Transition}},
  author = {Yu-Tian Li},
  howpublished = {arXiv},
  year = {2026},
  month = aug,
  doi = {10.48550/arxiv.2608.30486},
  eprint = {2608.30486},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2608.30486},
}

RIS

TY  - UNPB
TI  - The Koornwinder--Kostenko--Teschl Conjecture for Jacobi Polynomials and the Discrete Laguerre Phase Transition
AU  - Yu-Tian Li
PY  - 2026
DA  - 2026/08/31
PB  - arXiv
DO  - 10.48550/arxiv.2608.30486
UR  - https://arxiv.org/abs/2608.30486
AB  - We prove the refined Koornwinder--Kostenko--Teschl conjecture. For the normalized weighted Jacobi function and all n∈ℕ_0, α,β≥0, and −1≤ x≤1, |g_n^(α,β)(x)| ≤ [ ((n+1)(n+α+β+1))/((n+α+1)(n+β+1)) ]^1/4≤1. 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≤0, a complementary one-sided Jacobi inequality gives the exact norm |e^−i tH_α|_ℓ^1→ℓ^∞ =(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.
ER  -

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