3 case, as we said, W$_{K}$ does not depend on temperature) and analyzed the T dependence of W ( ω$_{c}$ ) 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 W$_{K}$ 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, Δ W ( ω$_{c}$ ) 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 Δ W$_{K}$ = 0, previous calculations 21 show that Δ W ( ω$_{c}$ ) changes sign at some ω$_{c}$ , becomes negative at larger ω$_{c}$ and approaches zero from a negative side. The frequency when Δ W ( ω$_{c}$ ) changes sign is of order Δ at weak coupling, but increases as the coupling increases, and at large coupling becomes comparable to a bandwidth ( ∼ 1 eV ). At such frequencies the approximation of a DOS by a constant is question- able at best, and the behavior of Δ W ( ω$_{c}$ ) should gen- erally be influenced by a nonzero Δ W$_{K}$ . In particular, the optical integral can either remain positive for all fre- quencies below interband transitions (for large enough positive Δ W$_{K}$ ), or change sign and remain negative (for negative Δ W$_{K}$ ). The first behavior would be consistent with Refs. 8,9, while the second would be consistent with Ref. 10. Δ W can even show more exotic behavior with more than one sign change (for a small positive Δ W$_{K}$ ). We show various cases schematically in Fig.1. integral in the SCS at T = 0 and in the NS extrapolated to T = 0 and compare the cut off effect Δ f ( ω$_{c}$ ) to Δ W$_{K}$ term. We also analyze the sign of Δ W ( ω$_{c}$ ) at large fre- quencies and discuss under what conditions theoretical W ( ∞ ) increases in the SCS. We perform calculations for four models. First is a conventional BCS model with impurities (BCSI model). Second is an Einstein boson (EB) model of fermions in- teracting with a single Einstein boson whose propaga- tor does not change between NS and SCS. These two cases will illustrate a conventional idea of the spectral weight in SCS being less than in NS. Then we con- sider two more sophisticated models: a phenomenological “marginal Fermi liquid with impurities” (MFLI) model of Norman and P´epin $^{30}$, and a microscopic collective bo- son (CB) model 31 in which in the NS fermions interact with a gapless continuum of bosonic excitations, but in a d − wave SCS a gapless continuum splits into a resonance and a gaped continuum. This model describes, in par- ticular, interaction of fermions with their own collective spin fluctuations 32 via where g is the spin-fermion coupling, and χ ( q, ω ) is the spin susceptibility whose dynamics changes between NS and SCS. From our analysis we found that the introduction of a finite fermionic bandwidth by means of a lattice has generally a notable effect on both W and Δ W . We found that for all models except for BCSI model, only 70% − 80% of the optical spectral weight is obtained by integrating up to the bandwidth. In these three models, there also exists a wide range of ω$_{c}$ in which the behavior of Δ W ( ω$_{c}$ ) is due to variation of Δ f ( ω$_{c}$ ) which is domi- nant comparable to the Δ W$_{K}$ term. This dominance of the cut off term is consistent with the analysis in Refs. 21,22,33. We also found that for all models except for the origi- nal version of the MFLI model the optical weight at the highest frequencies is greater in the NS than in the SCS (i.e., Δ W < 0). This observation is consistent with the findings of Abanov and Chubukov $^{32}$, Benfatto et. al. $^{28}$, and Karakozov and Maksimov $^{34}$. In the original ver- sion of the MFLI model 30 the spectral weight in SCS was found to be greater than in the NS (Δ W > 0). We show that the behavior of Δ W ( ω$_{c}$ ) in this model cru- cially depends on how the fermionic self-energy modeled to fit ARPES data in a NS is modified when a system becomes a superconductor and can be of either sign. We also found, however, that ω$_{c}$ at which Δ W becomes neg- ative rapidly increases with the coupling strength and at strong coupling becomes comparable to the bandwidth. In the CB model, which, we believe, is most appropriate for the application to the cuprates, Δ W$_{K}$ = Δ W ( ∞ ) is quite small, and at strong coupling a negative Δ W ( ω$_{c}$ ) up to ω$_{c}$ ∼ 1 eV is nearly compensated by the optical integral between ω$_{c}$ and “infinity”, which, in practice, is In our work, we perform direct numerical calculations of optical integrals at T = 0 for a lattice dispersion ex- tracted from ARPES of the cuprates. The goal of our work is two-fold. First, we perform calculations of the optical integral in the NS and analyze how rapidly W ( ω$_{c}$ ) approaches W$_{K}$ , in other words we check how much of the Kubo sum is recovered up to the scale of the band- width. Second, we analyze the difference between optical FIG. 1: Schematic behavior of Δ W vs ω c , Eq. (4). The limiting value of Δ W at ω c = ∞ is Δ W K given by Eq. (3) Depending on the value of Δ W K , there can be either one sign change of Δ W (panels a and c), or no sign changes (panel b), or two sign changes (panel d). Σ( k, Ω) = 3 g 2 ∫ dω 2 π d $^{2}$q (2 π ) 2 χ ( q, ω ) G ( k + q, ω + Ω) (6) 0 c  W 0 c  W 0 c  W 0 c  W (a) (b) (c) (d)