We consider the differential sum rule for the effective scattering rate $1/\tau \left(\omega \right)$ and optical conductivity ${\sigma }_1\left(\omega \right)$ in a dirty BCS superconductor, for arbitrary ratio of the superconducting gap $\Delta$ and the normal state constant damping rate $1/\tau .$ We show that if $\tau$ is independent of T, the area under $1/\tau \left(\omega \right)$ does not change between the normal and the superconducting states, i.e., there exists an exact differential sum rule for the scattering rate. For any value of the dimensionless parameter $\Delta \tau ,$ the sum rule is exhausted at frequencies controlled by $\Delta .$ We show that in the dirty limit the convergence of the differential sum rule for the scattering rate is much faster then the convergence of the f-sum rule, but slower then the convergence of the differential sum rule for conductivity.

• Received 16 December 2002

DOI:https://doi.org/10.1103/PhysRevB.68.024504