Sum rules and the collective oscillations of a quantum gas

Abstract
The formalism of sum rules is developed and applied to investigate the collective oscillations and the correlation effects exhibited by interacting quantum gases, with special emphasis on the role of dimensionality. General inequalities are derived and used to obtain rigorous results on the behaviour of long range order. Fully analytic expressions for the density response function are shown in some relevant cases, including the Bogoliubov and the Tonks-Girardeau gases. Applications are presented in the presence of harmonic trapping as well as of optical lattices.