The pairing Hamiltonian with two pairs in many levels

We address the pairing problem for two pairs of fermions in many levels of a potential well. We introduce a new number, referred to as ‘like-seniority’, which provides a measure of the degree of collectivity of the states. We point out that two critical values of the coupling constant might exist and we explore the behaviour of the energy, the wave functions and the transition amplitudes, in particular around these critical points. It is worth noticing that, in some definite cases, including the one of metal bands, the pairs can escape the trapping mechanism.