  We investigate a scale-invariant two-component Fermi gas in a time-dependent
isotropic harmonic potential. The exact time evolution of the density
distribution in position space in any spatial dimension is obtained. Two
experimentally relevant examples, an abrupt change and a periodic modulation of
the trapping frequency are solved. Small deviations from scale invariance and
isotropy of the confinement are addressed within first order perturbation
theory. We discuss the consequences for experiments with ultracold quantum
gases such as the excitation of a tower of undamped breathing modes and a new
alternative for measuring the Tan contact.
