Energies of the ground state and first excited $0^{+}$ state in an   exactly solvable pairing model
  Several approximations are tested by calculating the ground-state energy and the energy of the first excited $0^{+}$ state using an exactly solvable model with two symmetric levels interacting via a pairing force. They are the BCS approximation (BCS), Lipkin - Nogami (LN) method, random-phase approximation (RPA), quasiparticle RPA (QRPA), the renormalized RPA (RRPA), and renormalized QRPA (RQRPA). It is shown that, in the strong-coupling regime, the QRPA which neglects the scattering term of the model Hamiltonian offers the best fit to the exact solutions. A recipe is proposed using the RRPA and RQRPA in combination with the pairing gap given by the LN method. Applying this recipe, it is shown that the normal-superfluid phase transition is avoided, and a reasonably good description for both of the ground-state energy and the energy of the first excited $0^{+}$ state is achieved. 
Nguyen Dinh Dang