text: In this work we analyzed the behavior of optical in- 
  ωc 
  o σ(ω)dω and Kubo sum rules in 
  tegrals W (ωc) ∝ 
  the normal and superconducting states of interacting 
  fermionic systems on a lattice. Our key goal was to 
  understand what sets the sign of ∆WK = ∆W (∞) be- 
  tween the normal and superconducting states and what 
  is the behavior of W (ωc) and ∆W (ωc) at ﬁnite ωc. In a 
  weak coupling BCS superconductor, ∆W (ωc) is positive 
  at ωc < 2∆ due to a contribution from superﬂuid den- 
  sity, but becomes negative at larger ωc, and approach a 
  negative value of ∆WK. Our study was motivated by fas- 
  cinating optical experiments on the cuprates7–10. In over- 
  doped cuprates, there is clear indication11 that ∆W (ωc) 
  becomes negative above a few ∆, consistent with BCS 
  behavior. In underdoped cuprates, two groups argued8,9 
  that ∆W integrated up to the bandwidth remains posi- 
  tive, while the other group argued10 that it is negative. 
  The reasoning why ∆WK may potentially change sign 
  at strong coupling involves the correlation between −WK 
  and the kinetic energy. 
  In the BCS limit, kinetic en- 
  ergy obviously increases in a SCS because of gap opening, 
  hence −WK increases, and ∆WK is negative. At strong 
  coupling, there is a counter eﬀect – fermions become more 
  mobile in a SCS due to a smaller self-energy. 
  
  R 