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

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