Methods of solving the system of equations for the energy gap in the revisited BCS theory of superconductivity

Abstract

The Bardeen-Cooper-Schrieffer (BCS) theory of superconductivity has been revisited in a series of papers [1] , [2] , [3] and in this context the equation for the energy gap was generalized to a system of integral equations. The physical consequences of this change are major, leading not only to the change of the critical temperature and of energy gap, but even to a change of the order of the phase transition and to multiple solutions for the energy gap. Nevertheless, finding the solutions of the proposed system of equations is much more complicated than solving the typical BCS gap equation and requires a careful analysis. This analysis is done here and consists of the following steps: writing the system of equations at finite temperature ( comparable with the energy gap ) and in the low temperature limit ( ); separate analysis of the equations and of their solutions in the two temperature ranges, (first) and (second) comparable with ; presenting the methods to consistenlty searching the solutions.