An Existence Theorem for the BCS Integral Equation

Abstract

An important consequence of the Bardeen-Cooper-Schrieffer (BCS) theory of superconductivity is the existence of a gap in the energy spectrum. In the original BCS paper 1 the gap function was found by solving a homogeneous nonlinear integral equation in which the phonon interaction energy was taken to be a constant in some appropriate region and to be zero everywhere else. In that case the gap function is a constant which is determined by a simple algebraic relation. If one considers more general interactions, the existence of a nonzero energy-gap function is then by no means obvious. We must mention, however, that certain numerical calculations employing linearization procedures do produce nontrivial solutions for the cases they consider. (See Ref. 2, where other references are cited).