Analysis of the Strong Coupling Limit of the Richardson Hamiltonian using the Dyson Mapping

The Richardson Hamiltonian describes superconducting correlations in a metallic nanograin. We do a perturbative analysis of this and related Hamiltonians, around the strong pairing limit, without having to invoke Bethe Ansatz solvability. Rather we make use of a boson expansion method known as the Dyson mapping. Thus we uncover a selection rule that facilitates both time-independent and time-dependent perturbation expansions. In principle the model we analise is realised in a very small metalic grain of a very regular shape. The results we obtain point to subtleties sometimes neglected when thinking of the superconducting state as a Bose-Einstein condensate. An appendix contains a general presentation of time-independent perturbation theory for operators with degenerate spectra, with recursive formulas for corrections of arbitrarily high orders.