Concentration and Non-Concentration for the Schrödinger Evolution on Zoll Manifolds

Abstract

We study the long time dynamics of the Schrodinger equation on Zoll manifolds. We establish criteria under which the solutions of the Schrodinger 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 Schrodinger operators: we prove that adding a potential \({V \in C^{\infty} (\mathbb{S}^{d})}\) to the Laplacian on the sphere results in the existence of geodesics \({\gamma}\) such that the uniform measure supported on \({\gamma}\) cannot be obtained as a weak-\({\star}\) accumulation point of the densities \({(|\psi_{n}|^{2} {vol}_{\mathbb{S}^d})}\) for any sequence of eigenfunctions \({(\psi_n)}\) of \({\Delta_{\mathbb{S}^{d}} - V}\). We also show that the same phenomenon occurs for the free Laplacian on certain Zoll surfaces.