Spectral properties of mesons in hot and dense matter from energy weighted sum rules

Energy weighted sum rules for the spectral function of mesons are derived, both in cold nuclear matter and at finite temperature, by matching the Dyson propagator with its spectral Lehmann representation at low and high energies. We illustrate our results by calculating the sum rules for specific models of the kaon and pion self-energy in a nuclear medium. We find satisfactory results from the lower energy weight sum rules, which reflect the contributions from the different quasi-particle and collective modes of the meson spectral function, and show examples of the sensitivity of the sum rules to model approximations. Finally, we discuss applications in asymmetric nuclear matter and in the study of the spectral properties of vector mesons.