14 modified MFLI models. It is interesting that this holds despite the fact that for large λ CB model displays the physics one apparently needs to reverse the sign of Δ W$_{K}$ – the absence of the quasiparticle peak in the NS and its emergence in the SCS accompanied by the dip and the hump at larger energies. The absence of coherent quasi- particle in the NS at large λ is also apparent form Fig 21 where we show the normal state distribution functions for two different λ . For large λ the jump (which indicates the presence of quasiparticles) virtually disappears. On a more careful look, we found that indifference of δW ( ω$_{c}$ ) to the increase of λ is merely the consequence of the fact that above we kept λω$_{sf}$ constant. Indeed, at small frequencies, fermionic self-energy in the NS is Σ ′ = λω , Σ” = λ $^{2}$ω$^{2}$/ ( λω$_{sf}$ ), and both Σ ′ and Σ ′′ increase with λ if we keep λω$_{sf}$ constant. But at frequencies larger than ω$_{sf}$ , which we actually probe by Δ W ( ω$_{c}$ ), the self- energy essentially depends only on λω$_{sf}$ , and increasing λ but keeping λω$_{sf}$ constant does not bring us closer to the physics associated with the recovery of electron coherence in the SCS. To detect this physics, we need to see how things evolve when we increase λω$_{sf}$ above the scale of Δ , i.e., consider a truly strong coupling when not only λ ≫ 1 but also the normal state Σ$_{NS}$( ω ≥ Δ) >> Δ. To address this issue, we took a larger λ for the same ω$_{sf}$ and re-did the calculation of the conductivities and optical integrals. The results for σ ( ω ) and Δ W ( ω$_{c}$ ) are presented in Fig. 22. We found the same behavior as be- fore, i.e., Δ W$_{K}$ is negative. But we also found that the larger is the overall scale for the self-energy, the larger is a frequency of zero-crossing of Δ W ( ω$_{c}$ ). In particular, for the same λ and ω$_{sf}$ that were used in Ref. 33 to fit the NS conductivity data, the zero crossing is at ∼ 0 . 8 eV which is quite close to the bandwidth. This implies that at a truly strong coupling the frequency at which Δ W ( ω$_{c}$ ) changes sign can well be larger than the bandwidth of 1 eV in which case Δ W integrated up to the bandwidth does indeed remain positive. Such behavior would be consistent with Refs.8,9. we also see from Fig. 22 that Δ W$_{K}$ becomes small at a truly strong coupling, and over a wide range of frequencies the behavior of Δ W ( ω$_{c}$ ) is predominantly governed by Δ f ( ω$_{c}$ ), i.e. by the cut-off term. 50 The implication is that, to first approximation, Δ W$_{K}$ can be neglected and positive Δ W ( w$_{c}$ ) integrated to a frequency where it is still positive is almost compen- sated by the integral over larger frequencies. This again would be consistent with the experimental data in Refs. 8,9. It is also instructive to understand the interplay be- tween the behavior of Δ W ( ω$_{c}$ ) and the behavior of the difference of the kinetic energy between the SCS and the NS, δ$_{KE}$ . We computed the kinetic energy as a function of λω$_{sf}$ and present the results in Fig. 23 for λ = 1 and 10. For a relatively weak λ = 1 the behavior is clearly BCS like- δ$_{KE}$ > 0 and increases with increasing λω$_{sf}$ . However, at large λ = 10, we see that the kinetic energy begin decreasing at large λω$_{sf}$ and eventually changes sign. The behavior of δ$_{KE}$ at a truly strong coupling is The result most relevant to the experiments on the cuprates is obtained for the spin fluctuation model. We found that at strong coupling, the zero-crossing of δW ( ω$_{c}$ ) occurs at a frequency which increases with the coupling strength and may become larger than the band- width at a truly strong coupling. Still, at even larger frequencies, Δ W ( ω$_{c}$ ) is negative. We considered four models: a BCS model with impu- rities, a model of fermions interacting with an Einstein boson, a phenomenological MFL model with impurities, and a model of fermions interacting with collective spin fluctuations. In all cases, we found that Δ W$_{K}$ is neg- ative, but how it evolves with ω$_{c}$ and how much of the sum rule is recovered by integrating up to the bandwidth depends on the model. The reasoning why Δ W$_{K}$ may potentially change sign at strong coupling involves the correlation between − W$_{K}$ and the kinetic energy. In the BCS limit, kinetic en- ergy obviously increases in a SCS because of gap opening, hence − W$_{K}$ increases, and Δ W$_{K}$ is negative. At strong coupling, there is a counter effect – fermions become more mobile in a SCS due to a smaller self-energy. In this work we analyzed the behavior of optical in- tegrals W ( ω$_{c}$ ) ∝ ∫ ω$_{c}$ o σ ( ω ) dω and Kubo sum rules in the normal and superconducting states of interacting fermionic systems on a lattice. Our key goal was to understand what sets the sign of Δ W$_{K}$ = Δ W ( ∞ ) be- tween the normal and superconducting states and what is the behavior of W ( ω$_{c}$ ) and Δ W ( ω$_{c}$ ) at finite ω$_{c}$ . In a weak coupling BCS superconductor, Δ W ( ω$_{c}$ ) is positive at ω$_{c}$ < 2Δ due to a contribution from superfluid den- sity, but becomes negative at larger ω$_{c}$ , and approach a negative value of Δ W$_{K}$ . Our study was motivated by fas- cinating optical experiments on the cuprates $^{7–10}$. In over- doped cuprates, there is clear indication 11 that Δ W ( ω$_{c}$ ) becomes negative above a few Δ, consistent with BCS behavior. In underdoped cuprates, two groups argued 8,9 that Δ W integrated up to the bandwidth remains posi- tive, while the other group argued 10 that it is negative. III. CONCLUSION We clearly see that the increase of the zero crossing frequency of Δ W ( ω$_{c}$ ) at a truly strong coupling is cor- related with the non-BCS behavior of δ$_{KE}$ . At the same time, the behavior of δW ( ω$_{c}$ ) is obviously not driven by the kinetic energy as eventually δW ( ω$_{c}$ ) changes sign and become negative. Rather, the increase in the frequency range where Δ W ( ω$_{c}$ ) remains positive and non-BCS be- havior of δ$_{KE}$ are two indications of the same effect that fermions are incoherent in the NS but acquire coherence in the SCS. consistent with earlier calculation of the kinetic energy for Ornstein-Zernike form of the spin susceptibility $^{43}$.