Ground-State Energy and Green's Function for Reduced Hamiltonian for Superconductivity

In their theory of superconductivity, Bardeen, Cooper, and Schrieffer made use of a reduced Hamiltonian which included only scattering of pairs of particles of opposite momentum and spin. It is shown that the solution they obtained by a variational method is correct to $O\left(\frac{1}{n}\right)$ for a large system. The single particle Green's function is derived and used to calculate the interaction energy.