text: approximation is that the self-energy can be computed 
  analytically. The full self-energy obtained with the lat- 
  tice dispersion is more involved and can only be obtained 
  numerically, but its structure is quite similar to the one 
  obtained with a constant DOS. 
  The self-energy for a constant DOS is given by 
  
  i 
  2π 
  (13) 
  
  Z 
  where 
  
  ω2 
  0 
  ω2 
  0 − (iΩ)2 
  
  and λn is a dimensionless electron-boson coupling. Inte- 
  grating and transforming to real frequencies, we obtain 
  
  π 
  2 
  
  We next consider the case of electrons interacting with 
  a single boson mode which by itself is not aﬀected by su- 
  perconductivity. The primary candidate for such mode is 
  an optical phonon. The imaginary part of the NS self en- 
  ergy has been discussed numerous times in the literature. 
  We make one simplifying assumption – approximate the 
  DOS by a constant in calculating fermionic self-energy. 
  We will, however, keep the full lattice dispersion in the 
  calculations of the optical integral. The advantage of this 
  (ω + ωo)2 − ∆2 ! 