propagation of singularities & the Schrodinger equation

Question

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?

Answer 1

There are propagation of singularities results for the Schr\"odinger operator, but they are usually stated on asymptotically Euclidean manifolds. Early results appear in a paper in CMP by Zelditch in around 82 or 83 exhibit the typical weird behaviour where singularities disappear then reappear at later discrete points of time. A more general `true' propagation theorem was proved by Craig, Kappeler and Strauss, but a much sharper set of results was eventually proved by Jared Wunsch, see his papers from the late '90's. I am not sure what the best results are on compact manifolds at this point, though there are some. For the sphere one can see the same phenomenon of solutions being singular at time 0 and then becoming smooth then instantaneously singular again at time $2\pi$, $4\pi$, and so on.