Observability Results Related to Fractional Schrödinger Operators

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