Energy gap in superconductors with singular asymmetric density of states

The ratio R=2${\mathrm{\Delta }}_0$/${\mathit{kT}}_{\mathit{c}}$ is calculated for δ-function singularities in the density of states. Both symmetric and asymmetric densities of states are considered. In accordance with the symmetric powerlike singularities [D.C. Mattis and M. Molina, Phys. Rev. B 44, 12 565 (1991)], it is shown that neither the symmetric nor the asymmetric singularities in the density of states can explain those experimentally observed values in high-${\mathit{T}}_{\mathit{c}}$ superconductors that are much higher than the BCS value of 3.52 in the weak-coupling limit.