Sum rules and electrodynamics of high-Tc cuprates in the pseudogap state

We explore connections between the electronic density of states (DOS) in a conducting system and the frequency dependence of the scattering rate $1/\tau(\omega)$ inferred from infrared spectroscopy. We show that changes in the DOS upon the development of energy gaps can be reliably tracked through the examination of the $1/\tau(\omega)$ spectra using the sum rules discussed in the text. Applying this analysis to the charge dynamics in high-$T_c$ cuprates we found radically different trends in the evolution of the DOS in the pseudogap state and in the superconducting state.