Closed-form expressions for the norm and many-body-operator matrix elements between exact-particle-number pairing wave functions (N-pair condensates with broken pairs), for both even and odd particle numbers. Whereas BCS and Hartree-Fock-Bogoliubov mean-field descriptions break particle-number (U(1)) symmetry, this formalism works directly with number-conserved states, which matters for finite systems like atomic nuclei and ultrasmall superconducting grains. Using properties of the generalized Kronecker delta, the otherwise combinatorial sums reduce to compact sums of MINORS and PFAFFIANS of submatrices formed from the pair coefficients and the Wigner-D-rotated canonical basis: one form comes directly from the generalized Kronecker delta, an equivalent one rewrites the Cooper-pair operator through an antisymmetric matrix P = U^T V U so overlaps become Pfaffians (paralleling Robledo's Pfaffian cure for the HFB sign problem). The result covers configuration mixing between different pair condensates (the generator coordinate method) and symmetry restoration by angular-momentum projection, reducing the computational cost of number-conserved pairing calculations.
This nuclear-theory paper (G. J. Fu) derives closed-form formulas for norm overlaps and many-body-operator matrix elements between wave functions that carry an EXACT particle number, within pairing t…