A differential equation for a class of correlation kernels

TL;DR

This paper derives a new differential equation for a correlation kernel object that generalizes known equations for the Schrödinger resolvent, with potential applications in random matrix theory and physics.

Contribution

A novel differential equation for the correlation kernel that extends the Gel'fand-Dikii equation to a broader class of models.

Findings

Equation reduces to Gel'fand-Dikii for diagonal case

Analytical and numerical exploration of special cases

Potential applications in random matrix theory

Abstract

A new differential equation is derived for an object ${\widehat S}(E,E^\prime,x)$, which when integrated over the appropriate range in $x$, yields the kernel $K(E,E^\prime)$ with which $n$-point correlation functions can be computed in a wide class of models. When $E{=}E^\prime$, the equation reduces to the equation for the diagonal resolvent ${\widehat R}(E,x)$ of the Schr\"odinger Hamiltonian ${H}{=}{-}\hbar^2\partial_x^2{+}u(x)$ that is familiar from the classic work of Gel'fand and Dikii, and which appears in many areas of physics. This more general equation may also prove to be useful in a wide range of applications. Some special cases relevant to random matrix theory are explored using analytical and numerical methods.