Some exact results for the particle number projected BCS approach of the   isovector proton-neutron pairing
  The mean values of a many-body Hamiltonian including a proton-neutron pairing term and matrix elements of one-, two- and four-body operators within a basis of particle number projected BCS states, are analytically expressed in terms of a single function Q(N) depending on the number of particles, $N$. The function Q(N) is calculated using a recursion in $N$ in which the shells and the BCS angles are kept the same for any step of iteration. An illustrative example is numerically considered in a restricted single particle space. Some specific features for the standard BCS, the projection after variation approach as well as for the variation after projection formalism, are pointed out. 
A. A. Raduta, M. I. Krivoruchenko, Amand Faessler
nucl-th hep-ph