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.
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