text: the spectral weight under the δ-functional piece of the 
  conductivity in the superconducting state. 
  
  I.  INTRODUCTION 
  
  In practice, the integration up to an inﬁnite frequency 
  is hardly possible, and more relevant issue for practical 
  applications is whether a sum rule is satisﬁed, at least ap- 
  proximately, for a situation when there is a single electron 
  band which crosses the Fermi level and is well separated 
  from other bands. Kubo considered this case in the same 
  paper of 1957 and derived the expression for the “band”, 
  or Kubo sum rule 
  
  The analysis of sum rules for optical conductivity has a 
  long history. Kubo, in an extensive paper1 in 1957, used 
  a general formalism of a statistical theory of irreversible 
  processes to investigate the behavior of the conductivity 
  in electronic systems. For a system of interacting elec- 
  trons, he derived the expression for the integral of the real 
  part of a (complex) electric conductivity σ(Ω) and found 
  that it is independent on the nature of the interactions 
  and reduces to 
  ′ 
  ∞ 
  ‘ 
  
  πe2 
  2N 
  ∇2 
  ~kx 
  Re σ(Ω) dΩ = WK = 
  ε~k n~k 
  (3) 
  ∞ 
  (1) 
  0 
  Z 
  X~k 