Harmonic-Potential Theorem: Implications for Approximate Many-Body Theories

We consider interacting particles in an external harmonic potential. We extend the $B=0$ case of the generalized Kohn theorem, giving a "harmonic-potential theorem" (HPT), demonstrating rigid, arbitrary-amplitude, time-oscillatory Schr\"odinger transport of a many-body eigenfunction. We show analytically that the time-dependent local-density approximation (TDLDA) satisfies the HPT exactly. Other approximations, such as linearized TDLDA with frequency-dependent exchange correlation kernel and certain inhomogeneous hydrodynamic formalisms, do not. A simple modification permits such explicitly frequency-dependent local theories to satisfy the HPT, however.