Kernels and point processes associated with Whittaker functions

TL;DR

This paper explores the properties of Whittaker functions, their associated kernels, and Hankel operators, revealing connections to random matrix theory and Painlevé equations through explicit matrix and differential equation analyses.

Contribution

It establishes a link between Whittaker functions, Hankel operators, and random matrix kernels, and derives differential equations for Hankel matrix determinants.

Findings

Hankel operator $\Gamma_\varphi$ relates to Jacobi weight moments.

Determinants satisfy Painlevé $PV$ differential equation.

Whittaker kernel connects to random matrix theory.

Abstract

This article considers Whittaker's function $W_{\kappa ,\mu }$ where $\kappa$ is real and $\mu$ is real or purely imaginary. Then $\varphi (x)=x^{-\mu -1/2}W_{\kappa ,\mu }(x)$ arises as the scattering function of a continuous time linear system with state space $L^2(1/2, \infty )$ and input and output spaces ${\bf C}$. The Hankel operator $\Gamma_\varphi$ on $L^2(0, \infty )$ is expressed as a matrix with respect to the Laguerre basis and gives the Hankel matrix of moments of a Jacobi weight $w$. The operation of translating $\varphi$ is equivalent to multiplying $w$ by an exponential factor to give $w_\varepsilon$. The determinant of the Hankel matrix of moments of $w_\varepsilon$ satisfies the $\sigma$ form of Painlev\'e's transcendental differential equation $PV$. It is shown that $\Gamma_\varphi$ gives rise to the Whittaker kernel from random matrix theory, as studied by Borodin and Olshanski (Comm. Math. Phys. 211 (2000), 335--358).\par