6 FIG. 4: Top - a conductivity plot for the BCSI case in the presence of a lattice. The parameters are Δ = 30 meV , Γ = 3 . 5 meV . Bottom – the behavior of Kubo sums. Note that (a) the spectral weight in the NS is always greater in the SCS, (b) the spectral weight decreases with Γ, and (c) the difference between NS and SCS decreases as Γ increases. FIG. 5: The evolution of optical integral in NS(top) and SCS(bottom) for BCSI case. Plots are made for clean limit (solid lines, Γ = 3 . 5 meV ) and dirty limit (dashed lines, Γ = 150 meV ) for Δ = 30 meV . Observe that (a) W (0) = 0 in the NS, but has a non-zero value in the SCS because of the δ -function (this value decreases in the dirty limit), and (b) the flat region in the SCS is due to the fact that σ $^{′}$( ω ) = 0 for Ω < 2Δ. Also note that ∼ 90 − 95% of the spectral weight is recovered up to 1 eV little variation of Δ W ( ω$_{c}$ ) at above 0 . 1 − 0 . 3 eV what implies that for larger ω$_{c}$ , Δ W ( ω$_{c}$ ) ≈ Δ W$_{K}$ >> Δ f ( ω$_{c}$ ). To make this more quantitative, we compare in Fig. 6 Δ W ( ω$_{c}$ ) obtained for a constant DOS, when Δ W ( ω$_{c}$ ) = Δ f ( ω$_{c}$ ), and for the actual lattice dispersion, when Δ W ( ω$_{c}$ ) = Δ W$_{K}$ + Δ f ( ω$_{c}$ ). In the clean limit there is obviously little cutoff dependence beyond 0 . 1 eV , i.e., Δ f ( ω$_{c}$ ) is truly small, and the difference between the two cases is just Δ W$_{K}$ . In the dirty limit, the situation is similar, but there is obviously more variation with ω$_{c}$ , and Δ f ( ω$_{c}$ ) becomes truly small only above 0 . 3 eV . Note also that the position of the dip in Δ W ( ω$_{c}$ ) in the clean limit is at a larger ω$_{c}$ in the presence of the lattice than in a continuum. 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 In the SCS, we obtain for ω < 0 and λ$_{n}$ is a dimensionless electron-boson coupling. Inte- grating and transforming to real frequencies, we obtain π where 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 Σ( iω ) = − i 2 π λ$_{n}$ ∫ dϵ$_{k}$d ( i Ω) χ ( i Ω) G ( ϵ$_{k}$, iω + i Ω) (13) χ ( i Ω) = ω 2 0 ω 2 0 − ( i Ω) 2 (14) Σ ′′ ( ω ) = − 2 λ$_{n}$ω$_{o}$ Θ( | ω | − ω$_{o}$ ) Σ ′ ( ω ) = − 1 2 λ$_{n}$ω$_{o}$ log ∣ ∣$^{ω}$ + ω$_{o}$ ω − ω$_{o}$ ∣ (15) ω < 0 Σ ′′ ( ω ) = − π 2 λ$_{n}$ω$_{o}$ Re ( ω + ω$_{o}$ √ ( ω + ω$_{o}$ ) 2 − Δ 2 ) 0 0.5 1 0.5 1  in eV $_{(}$ω) Conductivities (BCSI) NS SC 2 0 50 100 160 180 200  in meV W K  in meV BCSI SC NS 0 0.5 1 0 0.5 1 $_{c}$ in eV W( c )/W( ) Normal State Optical Sum (BCSI) Dirty Limit Clean Limit 0 0.5 1 0 0.5 1 $_{c}$ in eV W( c )/W( ) Superconducting State Optical Sum (BCSI) Dirty Limit Clean Limit B. The Einstein boson model