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We calculate the $c$-axis kinetic energy difference between normal andsuperconducting state for coherent and for incoherent interlayer couplingbetween CuO$_{2}$ planes. For coherent coupling the ratio of the missingconductivity spectral weight to the superfluid density is equal to one andthere is no violation of the conventional sum rule, but for the incoherent casewe find it is always greater than one whatever the nature of the impuritypotential may be. To model more explicitly YBa$_{2}$Cu$_{3}$O$_{7-x}$ aroundoptimum doping, which is found to obey the sum rule, we consider a plane-chainmodel and show that the sum rule still applies. A violation of the sum rule ofeither sign is found even for coherent coupling when the in-plane density ofelectronic states depends on energy on a scale of the order of the gap.Comment: To appear in PRB Rapid Communicatio

Conductivity sum rule: Comparison of coherent and incoherentc-axis coupling