uc l - th / 0 40 40 42 v 2 7 J ul 2 00 4 Classification of states of single - j fermions with J - pairing interaction

In this paper we show that a system of three fermions is exactly solvable for the case of a single-j in the presence of an angular momentum-J pairing interaction. On the basis of the solutions for this system, we obtain new sum rules for six-j symbols. When the Hamiltonian contains only an interaction between pairs of fermions coupled to spin J = J max = 2j − 1, the " non-integer " eigenvalues of three fermions with angular momentum I around the maximum appear as " non-integer " eigenvalues of four fermions if I is around (or larger than) J max. This pattern is also found in five and six fermion systems. A boson system with spin l exhibits a similar pattern.