Strong Coupling Expansion for the Pairing Hamiltonian

The paper is devoted to the effects of superconducting pairing in small metallic grains. It turns out that at strong superconducting coupling and in the limit of large Thouless conductance one can explicitly determine the low energy spectrum of the problem. We start with the strong coupling limit and develop a systematic expansion in powers of the inverse coupling constant for the many-particle spectrum of the system. The strong coupling expansion is based on the formal exact solution of the Richardson model and converges for realistic values of the coupling constant. We use this expansion to study the low energy excitations of the system, in particular energy and spin gaps in the many-body spectrum.