Conductivity sum rule; comparison of coherent and incoherent c-axis coupling

We calculate the $c$-axis kinetic energy difference between normal and superconducting state for coherent and for incoherent interlayer coupling between CuO$_{2}$ planes. For coherent coupling the ratio of the missing conductivity spectral weight to the superfluid density is equal to one and there is no violation of the conventional sum rule, but for the incoherent case we find it is always greater than one whatever the nature of the impurity potential may be. To model more explicitly YBa$_{2}$Cu$_{3}$O$_{7-x}$ around optimum doping, which is found to obey the sum rule, we consider a plane-chain model and show that the sum rule still applies. A violation of the sum rule of either sign is found even for coherent coupling when the in-plane density of electronic states depends on energy on a scale of the order of the gap.