Hankel operators that commute with second-order differential operators

TL;DR

This paper characterizes Hankel operators that commute with specific second-order differential operators, especially those related to hypergeometric equations, and explores their applications in random matrix theory and operator decay properties.

Contribution

It provides a comprehensive classification of commuting Hankel and differential operators for quadratic, hyperbolic, and trigonometric functions, linking to hypergeometric equations.

Findings

Classification of commuting Hankel and differential operators.

Identification of cases relevant to random matrix theory.

Results on exponential decay of Hankel operator singular numbers.

Abstract

Suppose that $\Gamma$ is a continuous and self-adjoint Hankel operator on $L^2(0, \infty)$ and that $Lf=-(d/dx(a(x)df/dx))+b(x)f(x)$ with $a(0)=0$. If $a$ and $b$ are both quadratic, hyperbolic or trigonometric functions, and $\phi$ satisfies a suitable form of Gauss's hypergeometric equation, or the confluent hypergeometric equation, then $L\Gamma =\Gamma L$. The paper catalogues the commuting pairs $\Gamma$ and $L$, including important cases in random matrix theory. There are also results proving rapid decay of the singular numbers of Hankel integral operators with kernels that are analytic and of exponential decay in the right half plane.