Dispersion and controllability for the Schrödinger equation on negatively curved manifolds

We study the time-dependent Schrödinger equation ı ∂u ∂t = -1 2 u, on a compact Riemannian manifold on which the geodesic flow has the Anosov property.Using the notion of semiclassical measures, we prove various results related to the dispersive properties of the Schrödinger propagator, and to controllability.where Op h (a) is a h-pseudodifferential operator of principal symbol a (see [Dimassi and Sjöstrand 1999], or Appendix A for a brief reminder).This construction gives a description of a state in terms of position and impulsion variables.Throughout the paper, we will denote by U t:= e ıt /2 the quantum propagator.