Intraband Optical Spectral Weight in the presence of a van Hove   singularity: application to Bi$_2$Sr$_2$CaCu$_2$O$_{8+\delta}$
  The Kubo single band sum rule is used to determine the optical spectral weight of a tight binding band with further than nearest neighbour hopping. We find for a wide range of parameters and doping concentrations that the change due to superconductivity at low temperature can be either negative or positive. In contrast, the kinetic energy change is always negative. We use an ARPES determined tight binding parametrization of Bi$_2$Sr$_2$CaCu$_2$O$_{8+\delta}$ to investigate whether this can account for recent observations of a positive change in the spectral weight due to the onset of superconductivity. With this band structure we find that in the relevant doping regime a straightforward BCS calculation of the optical spectral weight cannot account for the experimental observations. 
F. Marsiglio, F. Carbone, A. Kuzmenko, D. van der Marel