text: in a given band is compensated by an appropriate change 
  of the spectral weight in other bands such that the total 
  spectral weight, integrated over all bands, is conserved, 
  as in Eq. 
  (1). Still, non-conservation of the spectral 
  weight within a given band is an interesting phenomenon 
  as the degree of non-conservation is an indicator of rele- 
  vant energy scales in the problem. Indeed, when relevant 
  energy scales are much smaller than the Fermi energy, 
  i.e., changes in the conductivity are conﬁned to a near 
  vicinity of a Fermi surface (FS), one can expand εk near 
  kF as εk = vF (k − kF ) + (k − kF )2/(2mB) + O(k − kF )3 
  and obtain ∇2 
  ε~k ≈ 1/mB [this approximation is equiv- 
  ~kx 
  alent to approximating the density of states (DOS) by a 
  constant]. Then WK becomes πne2/(2mB) which does 
  not depend on temperature. The scale of the tempera- 
  ture dependence of WK is then an indicator how far in 
  energy the changes in conductivity extend when, e.g., a 
  system evolves from a normal metal to a superconductor. 
  Because relevant energy scales increase with the interac- 
  tion strength, the temperature dependence of WK is also 
  an indirect indicator of whether a system is in a weak, 
  intermediate, or strong coupling regime. 