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 $n\in \mathbb{N}\_0$, $\alpha ,\beta \ge 0$, and $-1\le x\le 1$, $$|g\_n^{(\alpha ,\beta )}(x)|\le {\left[\frac{(n+1)(n+\alpha +\beta +1)}{(n+\alpha +1)(n+\beta +1)}\right]}^{1/4}\le 1.$$ 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 $-1<\alpha \le 0$, a complementary one-sided Jacobi inequality gives the exact norm $\Vert {\mathrm{e}}^{-\mathrm{i}t{H}_{\alpha }}{\Vert }_{{\ell }^1\to {\ell }^{\infty }}=(1+t^2{)}^{-(1+\alpha )/2}$. 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.

38 pages