A differential equation for a class of correlation kernels

A new differential equation is derived for an object [Formula: see text], which when integrated over the appropriate range in x, yields the kernel [Formula: see text] with which n-point correlation functions can be computed in a wide class of models. When [Formula: see text], the equation reduces to the equation for the diagonal resolvent [Formula: see text] of the Schrödinger Hamiltonian [Formula: see text] 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.