A differential equation for a class of correlation kernels

Clifford V. Johnson

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arXiv · 2505.10622

A differential equation for a class of correlation kernels

Clifford V. Johnson

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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.

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Clifford V. Johnson. 2025-05-15. A differential equation for a class of correlation kernels. https://arxiv.org/abs/2505.10622

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