A differential equation for a class of correlation kernels

Abstract

A new differential equation is derived for an object $\hat{S}(E,E^{\prime },x)$, 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 $\hat{R}(E,x)$ of the Schrödinger Hamiltonian $H=-{\hslash }^2{\partial }_x^2+u(x)$ 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.

5 pages, 3 figures