## ABSTRACT

An ultimate goal of the theory of superconductivity is to provide an expression for T_{c}
as a function of some well-defined parameters characterizing the material. In the framework of BCS theory, the Eliashberg equation (3.74) for the gap function properly takes into account a realistic phonon spectrum and retardation of the electron–phonon interaction. T_{c}
is fairly approximated by McMillan’s formula (3.96), which works well for simple metals and their alloys. But applying a theory of this kind to high-T_{c}
cuprates is problematic. Since bare electron bands are narrow, strong correlations result in the Mott insulating state of undoped parent compounds. As a result, μ^{*} and λ are ill defined in doped cuprates and polaronic effects are important as in many doped semiconductors [13]. Hence, an estimate of T_{c}
in cuprates within BCS theory appears to be an exercise in calculating μ^{*} rather than T_{c}
itself. Also, one cannot increase λ without accounting for a polaron collapse of the band (chapter 4). This appears at λ ≃ 1 for uncorrected electrons (holes) and even at a smaller value of bare electron–phonon coupling in strongly correlated models [216].