Sum Rules in Quantum Liquids

We review the linear response theory in the horizontal quantum liquids framework spanning from Coulomb liquids to Atomic gases. There are several well known references about this subject, like the onset of the Kramers-Kronig relations and the fluctuation-dissipation theorem. For the Coulomb systems we show the connection between the linear response function and the dielectric function which settles a parallelism between statistical mechanics and electrostatic properties. For (very degenerate, dilute, trapped, two dimensional) Atomic (Bose) gases we will prove, in full generality, how the response properties of the gas depend from the frequency of the harmonic trap.