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

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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 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.

Original languageEnglish
Pages (from-to)6299-6310
Number of pages12
JournalJournal of Physics A: Mathematical and General
Volume37
Issue number24
DOIs
StatePublished - 18 Jun 2004

ASJC Scopus subject areas

  • Statistical and Nonlinear Physics
  • Mathematical Physics
  • Physics and Astronomy (all)

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