Functional averages in the theory of superconductivity

The grand canonical partition function of a superconductor described byGorkov's model Hamiltonian is represented as a functional integral with Gaussian measure. The integrand can be regarded as the partition function of a free Fermi system which interacts with a fluctuating external source potential. Perturbation-theoretic techniques are applied to the latter partition function. TheGibbs' potential proves to be stationary with respect to the energy gap parameterΔ. From the stationarity condition an equation forΔ is obtained which is a generalization of the usual Bardeen-Cooper-Schrieffer (BCS) equation. For the evaluation of the functional integral a variational procedure is employed. It leads to an expression for theGibbs' potential which shows a further remarkable stationarity property. As its simplest approximation this expression contains a result that was firstly derived by Thouless in the ladder graph approximation.