text: case, as we said, WK does not depend on temperature) 
  and analyzed the T dependence of W (ωc) due to the T 
  dependence of the cut-oﬀ term. They found a good agree- 
  ment with the experiments. This still does not solve the 
  problem fully as amount of the T dependence of WK in 
  the same model but with a lattice dispersion has not been 
  analyzed. For a superconductor, which of the two terms 
  contributes more, remains an open issue. At small fre- 
  quencies, ∆W (ωc) between a SCS and a NS is positive 
  simply because σ(Ω) in a SCS has a δ−functional term. 
  In the models with a constant DOS, for which ∆WK = 0, 
  previous calculations21 show that ∆W (ωc) changes sign 
  at some ωc, becomes negative at larger ωc and approaches 
  zero from a negative side. The frequency when ∆W (ωc) 
  changes sign is of order ∆ at weak coupling, but increases 
  as the coupling increases, and at large coupling becomes 
  comparable to a bandwidth (∼ 1eV ). At such frequencies 
  the approximation of a DOS by a constant is question- 
  able at best, and the behavior of ∆W (ωc) should gen- 
  erally be inﬂuenced by a nonzero ∆WK . In particular, 
  the optical integral can either remain positive for all fre- 
  quencies below interband transitions (for large enough 
  positive ∆WK), or change sign and remain negative (for 
  negative ∆WK ). The ﬁrst behavior would be consistent 
  with Refs. 8,9, while the second would be consistent with 
  Ref. 10. ∆W can even show more exotic behavior with 
  more than one sign change (for a small positive ∆WK ). 
  We show various cases schematically in Fig.1. 