I've been thinking about the following propagation of singularities result:

Let $X$ be a compact manifold, and let $P$ be a differential operator (of, say, order $m$) on $X$ whose principal symbol $\sigma_m(P)$ is real-valued. Suppose that $Pu=0$. Then the wavefront set of the solution $u$ is a union of maximally extended (null) bicharacteristics of $\sigma_m(P)$ in the co-sphere bundle $S^*X$.

Let's consider the Schrodinger operator on $X\times\mathbb{R}$:

$P=-i\partial_t+\Delta_x$.

My question is what, if anything, does the above propagation theorem tell us about solutions $u$ to the homogeneous Schrodinger equation?