Concentration and non-concentration for the Schr\"odinger evolution on Zoll manifolds

We study the long time dynamics of the Schr\"odinger equation on Zoll manifolds. We establish criteria under which the solutions of the Schr\"odinger equation can or cannot concentrate on a given closed geodesic. As an application, we derive some results on the set of semiclassical measures for eigenfunctions of Schr\"odinger operators: we prove that adding a potential to the Laplacian on the sphere results on the existence of geodesics $\gamma$ such that $\delta_\gamma$ cannot be obtained as a semiclassical measure for some sequence of eigenfunctions. We also show that the same phenomenon occurs for the free Laplacian on certain Zoll surfaces.