Superconductivity in systems exhibiting the Altshuler-Aronov anomaly

Making use of generalized Eliashberg equations, we describe the Altshuler-Aronov (AA) effect and superconductivity on equal footing. We derive explicit expressions for the Coulomb pseudopotential in 3D, taking into account also the anomalous diffusion. We present a full numerical solution for two normal-state and two anomalous self-energies. In the normal state, we amend the known results for the purely electronic AA effect; with electron-phonon coupling turned on, we find additional anomalies in the density of states close to the phonon energy. We study how the critical temperature and density of states of strongly disordered 3D superconductors change with normal-state resistivity. We find that the type of transition from the superconducting to the insulating state depends on the strength of electron-phonon coupling: at weak coupling there exists an intermediate normal state, whereas at strong coupling the transition is direct.