  We establish observability inequalities for various problems involving
fractional Schr\"odinger operators $(-\Delta)^{\alpha/2}+V$, $\alpha>0$, on a
compact Riemannian manifold. Observability from an open set for the
corresponding fractional Schr\"odinger evolution equation with $\alpha>1$ is
proved to hold as soon as the observation set satisfies the Geometric Control
Condition; it is also shown that this condition is necessary when the manifold
is the $d$-dimensional sphere equipped with the standard metric. This is in
stark contrast with the case of eigenfunctions. We construct potentials on the
two-sphere with the property that there exist two points on the sphere such
that eigenfunctions of $-\Delta+V$ are uniformly observable from an arbitrarily
small neighborhood of those two points. This condition is much weaker than the
Geometric Control Condition, which is necessary for uniform observability of
eigenfunctions for the free Laplacian on the sphere. The same result also holds
for eigenfunctions of $(-\Delta)^{\alpha/2}+V$, for any $\alpha>0$.

Analysis of PDEs