Pairing correlations studied in the two-level model

We study the pairing energy in the ground state of a system of N particles occupying a symmetric two-level model space with a level degeneracy Ω. Assuming a pairing Hamiltonian, we calculate the energies of the projected BCS states, using the exact, the Kamlah, and the Lipkin-Nogami particle number projection methods. We find that conclusions regarding the quality of the BCS approximation as well as various approximate projection methods, drawn from studying the N=Ω case alone, are not valid in the more general N≠Ω case when the pairing interaction is weak.