Consistency of particle-particle random-phase approximation and its   renormalizations
  The consistency condition is tested within the particle-particle random-phase approximation (RPA), renormalized RPA (RRPA) and the self-consistent RPA (SCRPA) making use of the Richardson model of pairing. The two-particle separation energy is calculated in two ways, namely as the energy of the first addition mode, which adds two particles to a core with N particles, and as the energy of the first removal mode, which removes two particles from the (N+2) - particle system to get back to the same N-particle core. The corresponding transitions generated by the pairing operators are also calculated. It is found that the results obtained in these two ways of calculations are close to each other only at large values of particle number N and/or small interaction strength. At N below 10 for a given value of the interaction strength, the discrepancy between the results obtained in two ways of calculations within the SCRPA is much smaller than those given by the RPA and RRPA. 
Nguyen Dinh Dang