Observability for generalized Schrödinger equations and quantum limits on product manifolds

Overview & Synopsis

Given a closed product Riemannian manifold N = M × M equipped with the product Riemannian metric g = h + h , we explore the observability properties for the generalized Schrödinger equation i∂ t u = F (g)u, where g is the Laplace-Beltrami operator on N and F : [0, +∞) → [0, +∞) is an increasing function. In this note, we prove observability in finite time on any open subset ω satisfying the so-called Vertical Geometric Control Condition, stipulating that any vertical geodesic meets ω, under the additional assumption that the spectrum of F (g) satisfies a gap condition. A first consequence is that observability on ω for the Schrödinger equation is a strictly weaker property than the usual Geometric Control Condition on any product of spheres. A second consequence is that the Dirac measure along any geodesic of N is never a quantum limit.