The description of superconductivity in terms of dielectric response function

A critical temperatureT _c of a superconducting transition is calculated for a rather general form of the electron-electron interaction. It is shown that even if both the energy and momentum dependence of the interaction is included, the equation determiningT _c coincides formally with the corresponding equation of the BCS theory. The kernel of this equation is a smooth real function of its variables; it is expressed through ρ(k, E), the spectral density of the inverse dielectric function of the system. The expression forT _c is written in terms of ρ(k, E); this enables us to analyze the dependence of the critical temperature on the properties of the metal in a normal state. Some simple models illustrating the results are considered, and a discussion of the limits onT _c is given.