  We display an interesting sum rule for the dynamical thermal conductivity for
many standard models of condensed matter in terms of the expectation of a
thermal operator. We present the thermal operator for several model systems of
current interest, which enable an evaluation of the sum rule and the Lorentz
number, the thermo electric figure of merit as well as the thermopower at high
frequencies. As a by product, we present exact formulae for the T=0 chemical
potential $\mu(0)$ for charged many-body systems, including the Hubbard model,
in terms of expectation values of extensive operators. Simple estimates are
provided for the thermopower of an infinitely correlated band model on the
triangular lattice, modeling the physics of the sodium cobalt oxide system. The
present result goes beyond the Heikes Mott formula for the thermopower, and
contains an additional transport correction that is sensitive to the lattice
topology as well as the sign of hopping.
