Observability for generalized Schr{\"o}dinger equations and quantum limits on product manifolds

Given a closed product Riemannian manifold N = M x M equipped with the product Riemannian metric g = h + h , we explore the observability properties for the generalized Schr{\"o}dinger equation i$\partial$ t u = F (g)u, where g is the Laplace-Beltrami operator on N and F : [0, +$\infty$) $\rightarrow$ [0, +$\infty$) is an increasing function. In this note, we prove observability in finite time on any open subset $\omega$ satisfying the so-called Vertical Geometric Control Condition, stipulating that any vertical geodesic meets $\omega$, under the additional assumption that the spectrum of F (g) satisfies a gap condition. A first consequence is that observability on $\omega$ for the Schr{\"o}dinger 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.