Experimentally observed extended saddle point singularity in the energy spectrum of YBa₂Cu₃O_6.9 and YBa₂Cu₄O₈ and some of the consequences

Angle-resolved photoemission experiments show that the high temperature superconductors YBa₂Cu₃O_6.9 and YBa₂Cu₄O₈ have a spectral density which exhibits an "extended" saddle point singularity along the Γ-Y symmetry direction, centered on the Y point. This, in turn, leads to a van Hove singularity in the density of states with a divergence stronger than the well-known logarithmic one. We examine the consequences of this singularity in several limiting cases on the superconducting transition temperature, the ratio 2Δ(0)/T_c, and the isotope effect. We conclude that this singularity alone, although it can possibly lead to sufficiently high transition temperatures and a vanishingly small isotope effect, does not explain the ratio of 2Δ(0)/T_c∼6 observed experimentally.