Strong-coupling properties of unbalanced Eliashberg superconductors

In this paper we investigate the thermodynamical properties of ``unbalanced'' superconductors, namely, systems where the electron-boson coupling $\lambda$ is different in the self-energy and in the Cooper channels. This situation is encountered in a variety of situation, as for instance in d-wave superconductors. Quite interesting is the case where the pairing in the self-energy is smaller than the one in the gap equation. In this case we predict a finite critical value $\lambda_c$ where the superconducting critical temperature $T_c$ diverges but the zero temperature gap is still finite. The specific heat, magnetic critical field and the penetration depth are also evaluated.