Fredholm's Minors of Arbitrary Order: Their Representations as a Determinant of Resolvents and in Terms of Free Fermions and an Explicit Formula for Their Functional Derivative

TL;DR

This paper introduces new linear recursion relations for Fredholm minors, representing them as determinants of resolvents and relating them to free fermions, along with an explicit formula for their functional derivatives.

Contribution

It provides novel recursion relations, a determinant representation of minors, a fermionic path integral interpretation, and an explicit derivative formula for Fredholm minors.

Findings

Recursion relations for Fredholm minors

Determinant representation of minors as resolvents

Explicit formula for functional derivatives

Abstract

We study the Fredholm minors associated with a Fredholm equation of the second type. We present a couple of new linear recursion relations involving the $n$th and $n-1$th minors, whose solution is a representation of the $n$th minor as an $n\times n$ determinant of resolvents. The latter is given a simple interpretation in terms of a path integral over non-interacting fermions. We also provide an explicit formula for the functional derivative of a Fredholm minor of order $n$ with respect to the kernel. Our formula is a linear combination of the $n$th and the $n\pm 1$th minors.