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 $n\in\mathbb N\_0$, $\alpha,\beta\ge0$, and $-1\le x\le1$, $$ |g\_n^{(\alpha,\beta)}(x)| \le \left[ \frac{(n+1)(n+\alpha+\beta+1)} {(n+\alpha+1)(n+\beta+1)} \right]^{1/4}\le1. $$ 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 $-1<\alpha\le0$, a complementary one-sided Jacobi inequality gives the exact norm $\|\mathrm{e}^{-\mathrm{i} tH_\alpha}\|_{\ell^1\to\ell^\infty} =(1+t^2)^{-(1+\alpha)/2}$. Thus the large-time decay exponent is $\min\{1,1+\alpha\}$ for every $\alpha>-1$. Bessel, Laguerre, and Darboux scaling limits show that the temporal and fixed-diagonal exponents are optimal.