Optical sum rule in metals with a strong interaction

The restricted optical sum rule and its dependence on the temperature, a superconducting gap and the cutoff energy have been investigated. As known this sum rule depends on the cutoff energy and the relaxation rate Γ(T) even for a homogeneous electron gas interacting with impurities or phonons. It is shown here that additional dependence of the spectral weight on a superconducting gap is very small in this model and this effect disappears totally when Γ=0. The model metal with a single band is considered in details. It is well known that for this model there is the dependence of the sum rule on the temperature and the energy gap even in the case when Γ=0. This dependence exists due to the smearing of the electron distribution function and it is expressed in the terms of Sommerfeld expansion. Here it is shown that these effects are considerably smaller than that of related with the relaxation rate if the band width is larger than the average phonon frequency. It is shown also that the experimental data about the temperature dependence of the spectral weight for the high-T_c materials can be successfully explained in the framework approach based on the temperature dependence of the relaxation rate.