Exact Treatment of a Charge Independent Pairing Interaction in a Degenerate Model

A charge independent pairing interaction in j-j coupling scheme and a charge-spin independent pairing interaction in L-S coupling scheme are exactly solved in a degenerate model by the aid of group theory. Groups used here are independent of the number of states concerned unlike the usual shell model technique. Our method is deeply connected with the generalized Bogoliubov transformation and it is conjectured that this transformation combined with projection into the definite number and the isospin (symmetry) state is very useful for the treatment of pair-interacting many body system. The classification of states through the group used here are equivalent to the seniority scheme generalized for unlike particles and degenerate levels. Our method simplifies mathematics somewhat and derives easily some new properties of seniority scheme for unlike particles.