  We consider Schr\"odinger equations with variable coefficients and the
harmonic potential. We suppose the perturbation is short-range type in the
sense of [Nakamura 2004]. We characterize the wave front set of the solutions
to the equation in terms of the classical scattering data and the propagator of
the unperturbed harmonic oscillator. In particular, we give a "recurrence of
singularities" type theorem for the propagation of the period $t=\pi$.
