The many levels pairing Hamiltonian for two pairs

We address the problem of two pairs of fermions living on an arbitrary number of single-particle levels of a potential well (mean field) and interacting through a pairing force in the framework of the Richardson equations. The associated solutions are classified in terms of a number vl, which reduces to the seniority v in the limit of a large pairing strength G and yields the number of pairs not developing a collective behaviour, their energy remaining finite in the G↦∞ limit. We express analytically, through the moments of the single-particle levels distribution, the collective mode energy and the two critical values G cr + and G cr - of the coupling which can exist on a single-particle level with no pair degeneracy. Notably G cr + and G cr -, when the number of single particle levels goes to infinity, merge into the critical coupling of a one-pair system G cr (when it exists), which is not envisioned by the Richardson theory. In correspondence of G cr, the system undergoes a transition from a mean-field- to a pairing-dominated regime. We finally explore the behaviour of the excitation energies, wave functions and pair transfer amplitudes versus G finding out that the former, for G > G cr -, come close to the BCS predictions, whereas the latter display a divergence at G cr, signaling the onset of a long-range off-diagonal order in the system