Invariance of the Kohn (sloshing) mode in a conserving theory

It is proven that the center of mass (COM or Kohn) oscillation of a many-body system in a harmonic trap coincides with the motion of a single particle as long as conserving approximations are applied to treat the interactions. The two conditions formulated by Kadanoff and Baym \cite{kb-book} are shown to be sufficient to preserve the COM mode. The result equally applies to zero and finite temperature, as well as to nonequilibrium situations, and to the linear and nonlinear response regimes.