Dependence of the critical temperature and energy gap of superconductors on a singularity in the density of states

It is shown how the critical temperature and zero-temperature energy gap decrease by shifting the position of the singularity in the density of states (DOS) with respect to the Fermi energy. The enhancement of the critical temperature due to the singularity (compared with the constant DOS near the Fermi surface) is not as large for strong-coupling superconductors as in the weak-coupling limit. Hence, the singularity in the DOS cannot be the exclusive reason for large values of critical temperatures in high-${\mathit{T}}_{\mathit{c}}$ superconductors.