  The third moment frequency sum rule for the density-density correlation
function is rederived in the presence of Umklapp processes. Upper and lower
bounds on the electron-electron Coulomb energy are derived in two-dimensional
and three-dimensional media, and the Umklapp processes are shown to be crucial
in determining the spectrum of the density fluctuations (especially for the
two-dimensional systems). This and other standard sum rules can be used in
conjunction with experimental spectroscopies (electron-energy loss
spectroscopy, optical ellipsometry, etc.) to analyse changes of the
electron-electron Coulomb energy at the superconducting transition in cuprates.

Superconductivity;Strongly Correlated Electrons