Exact results for the particle-number-projected BCS approach with 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 iteration step. An illustrative example is numerically considered in a restricted single-particle space. Some specific features of the standard BCS, the projection after variation approach, and the variation after projection formalism are pointed out.