The van Hove singularity and two-dimensional superconductivity. Exact analytical results

We investigate a weak-coupling approach to superconductivity in the density of states that develops in the two-dimensional lattice with the van Hove singularity located at the Fermi level. Exact analytical expressions for the superconducting order parameter at zero temperature (Delta(0)) and for discontinuity in the specific heat at critical temperature (DeltaC) were derived. We show that the presence of the logarithmic singularity hardly affects the ratio 2Delta(0)/kT(c), whereas the temperature dependence of DeltaC can pronouncedly deviate from the standard BCS result.