  We solve by Dynamical Mean Field Theory a toy-model which has a phase diagram
strikingly similar to that of high $T_c$ superconductors: a bell-shaped
superconducting region adjacent the Mott insulator and a normal phase that
evolves from a conventional Fermi liquid to a pseudogapped semi-metal as the
Mott transition is approached. Guided by the physics of the impurity model that
is self-consistently solved within Dynamical Mean Field Theory, we introduce an
analytical ansatz to model the dynamical behavior across the various phases
which fits very accurately the numerical data. The ansatz is based on the
assumption that the wave-function renormalization, that is very severe
especially in the pseudogap phase close to the Mott transition, is perfectly
canceled by the vertex corrections in the Cooper pairing channel.A remarkable
outcome is that a superconducting state can develop even from a pseudogapped
normal state, in which there are no low-energy quasiparticles. The overall
physical scenario that emerges, although unraveled in a specific model and in
an infinite-coordination Bethe lattice, can be interpreted in terms of so
general arguments to suggest that it can be realized in other correlated
systems.
