Solvable many-level pairing model in a boson formalism

We analyze an exactly solvable many-level pairing model (the picket-fence model) in a boson formalism. By making use of a Marumori-type mapping procedure, we construct a boson image of the model Hamiltonian. Only two- and four-boson operators are included in the boson expansion. We show that the physical spectrum of this Hamiltonian coincides with the exact one, and, in particular, we recover Richardson's equations. We apply this boson formalism to search for possible extensions of the random-phase approximation (RPA). We construct renormalized RPA-type equations in correspondence with different expansions of the boson operators. We discuss two extensions of the RPA and test them by comparing their predictions with the exact results and with those obtained within the RPA and the self-consistent RPA.