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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

Joshua Feinberg
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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.

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Joshua Feinberg. 2004-03-18. 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. https://doi.org/10.1088/0305-4470/37/24/008

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