Strong-coupling expansion for the pairing Hamiltonian for small superconducting metallic grains

The paper is devoted to the study of the effects due to superconducting pairing in small metallic grains. We explicitly determine the low-energy spectrum of the problem at strong superconducting coupling and in the limit of large Thouless conductance. 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.