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

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 nth and (n − 1)th minors, whose solution is a representation of the nth minor as an n × 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 nth and the (n ± 1)th minors.