Correlation energy of the pairing Hamiltonian

We study the correlation energy associated with the pair fluctuations in BCS theory. We use a schematic two-level pairing model and discuss the behavior of the correlation energy across shell closures, including the even-odd differences. It is shown that the random phase approximation (RPA) to the ground state energy reproduces the exact solutions quite well both in the normal fluid and in the superfluid phases. In marked contrast, other methods to improve the BCS energies such as Lipkin-Nogami method, only work when the strength of a pairing interaction is large. We thus conclude that the RPA approach is better for a systematic theory of nuclear binding energies.