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

TL;DR

This paper investigates the long-term behavior of Schr"odinger solutions on Zoll manifolds, identifying conditions for concentration or dispersion along geodesics, and explores implications for eigenfunction measures with added potentials.

Contribution

It establishes criteria for Schr"odinger concentration on geodesics and demonstrates how potentials influence semiclassical measures on Zoll manifolds.

Findings

Solutions can or cannot concentrate on specific geodesics based on established criteria.

Adding potentials to the Laplacian affects the set of possible semiclassical measures.

Certain Zoll surfaces exhibit phenomena similar to the sphere with respect to eigenfunction measures.

Abstract

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.