12 form of the imaginary part. FIG. 17: Conductivities and Δ W for a fixed λω$_{sf}$ . Top – ω$_{sf}$ = 26 meV , λ = 1, ω o = 40 meV , Z o = 0 . 77 Bottom – ω$_{sf}$ = 2 . 6 meV , λ = 10, ω o = 13 . 5 meV , Z o = 1 . 22. The zero crossing for Δ W is not affected by a change in λ because it is determined only by λω$_{sf}$ . We set Δ = 30 meV . FIG. 19: The evolution of the optical integrals in the NS and the SCS in the CB model. Note that about ∼ 75% of the spectral weight is recovered up to 1 eV . We set ω$_{sf}$ = 26 meV , λ = 1, and Δ = 30 meV . FIG. 18: The behavior of Kubo sums in the CB model. Note that the spectral weight in the NS is always larger than in the SCS. We set ω$_{sf}$ = 26 meV , λ = 1, and Δ = 30 meV . FIG. 20: Δ W (in meV) for λ = 1(top) and λ = 10(bottom). We used ω$_{sf}$ = 26 meV/λ and Δ = 30 meV . The zero crossing is not affected because we keep λω$_{sf}$ constant. The notable difference is the widening of the dip at a larger λ . We performed the same calculations of conductivities and optical integrals as in the previous three cases. The results are summarized in Figs. 17 - 22. Fig 17 shows con- ductivities in the NS and the SCS for two couplings λ = 1 and λ = 10 (keeping λω$_{sf}$ constant). Other parameters Z$_{o}$ and ω$_{o}$ are calculated according to the discussion after Eq 21. for ω$_{sf}$ = 26 meV , λ = 1, we find ω$_{o}$ = 40 meV , Z$_{o}$ = 0 . 77. And for ω$_{sf}$ = 2 . 6 meV , λ = 10, we find ω$_{o}$ = 13 . 5 meV , Z$_{o}$ = 1 . 22. Note that the conductivity in the SCS starts at 2Δ + ω$_{o}$ (i.e. the resonance energy 0 0.5 1 0 0.2 0.4  in eV $_{(}$ω) Conductivities (CB model   =1) NS SC 2 + o 0 0.5 1 0.2 0.6 1  in eV $_{(}$ω) Conductivities (CB model   =10) NS SC 2  +  o 0 0.5 1 0.2 0.6 1 $_{c}$ in eV W( c )/W( ) NS Optical Sums (CB model) 0 0.5 1 0.2 0.6 1 $_{c}$ in eV W( c )/W( ) SCS Optical Sums (CB model) 1 2 3 120 160 200  (coupling) CB model ( $_{o}$=40 meV) W K (meV) SCS NS 0 0.5 1 −20 0 20  $_{c}$ in eV W SC $_{(}$ω c ) − W NS $_{(}$ω c )  W (CB model  =1) with lattice without lattice  W$_{K}$ 0.2 0.6 1 −20 0 20  $_{c}$ in eV W SC $_{(}$ω c ) − W NS $_{(}$ω c )  W (CB model  =10) with lattice without lattice  W$_{K}$