  We study the pairing Hamiltonian in a set of non degenerate levels. First, we
review in the path integral framework the spontaneous breaking of the U(1)
symmetry occurring in such a system for the degenerate situation. Then the
behaviors with the coupling constant of the ground state energy in the
multilevel and in the degenerate case are compared. Next we discuss, in the
multilevel case, an exact strong coupling expansion for the ground state energy
which introduces the moments of the single particle level distribution. The
domain of validity of the expansion, which is known in the macroscopic limit,
is explored for finite systems and its implications for the energy of the
latter is discussed. Finally the seniority and Gaudin excitations of the
pairing Hamiltonian are addressed and shown to display the same gap in leading
order.
