Concentration and non concentration for the Schrödinger equation on Zoll manifolds

Fabricio Macià (1) , Gabriel Riviere (2)

1 UPM - Universidad Politécnica de Madrid
2 LPP - Laboratoire Paul Painlevé - UMR 8524

We study the long time dynamics of the Schrödinger equation on Zoll man-ifolds. We establish criteria under which the solutions of the Schrödinger 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 Schrödinger 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.

Mots clés: Semiclassical analysis; Integrable Hamiltonian systems; Semiclassical measures; Zoll manifolds; Schrödinger equation

Domaines: Equations aux dérivées partielles [math.AP]; Physique mathématique [math-ph]; Théorie spectrale [math.SP]

Fabricio Macià, Gabriel Riviere. Concentration and non concentration for the Schrödinger equation on Zoll manifolds. 2015. ⟨hal-01153285⟩