A differential equation for a class of correlation kernels

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 Schrödinger 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.

Figures (3)

• Figure 1: Imaginary part of the solution to Eq. (\ref{['eq:GD2']}) for ${\widehat{R}}(E,x)$, compared to $\psi(E,x)^2$, up to a normalization. The real part is also shown. Here $E{=}1$, and $\hbar{=}1$.
• Figure 2: Imaginary part of the solution for ${\widehat{S}}(E,E^\prime,x)$ to Eq. (\ref{['eq:beyondGD']}), compared to $\psi(E,x)\psi(E^\prime,x)$, up to a normalization. Here $E{=}1$, $E^\prime{=}1.001$, and $\hbar{=}1$.
• Figure 3: Imaginary part of the solution for ${\widehat{S}}(E,E^\prime,x)$ to Eq. (\ref{['eq:beyondGD']}), compared to $\psi(E,x)\psi(E^\prime,x)$, up to a normalization. Here $E{=}1$, $E^\prime{=}1.01$, and $\hbar{=}1$.