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We display an interesting sum rule for the dynamical thermal conductivity formany standard models of condensed matter in terms of the expectation of athermal operator. We present the thermal operator for several model systems ofcurrent interest, which enable an evaluation of the sum rule and the Lorentznumber, the thermo electric figure of merit as well as the thermopower at highfrequencies. As a by product, we present exact formulae for the T=0 chemicalpotential $\mu(0)$ for charged many-body systems, including the Hubbard model,in terms of expectation values of extensive operators. Simple estimates areprovided for the thermopower of an infinitely correlated band model on thetriangular lattice, modeling the physics of the sodium cobalt oxide system. Thepresent result goes beyond the Heikes Mott formula for the thermopower, andcontains an additional transport correction that is sensitive to the latticetopology as well as the sign of hopping.Comment: Erratum to Phys Rev incorporated, fixing typos in Eqs(41,87,88

Sum rule for thermal conductivity and dynamical thermal transport coefficients in condensed matter