Approximation of the Fractional Schrödinger Propagator on Compact Manifolds

Let £ be a second order positive, elliptic differential operator that is self-adjoint with respect to some C∞ density dx on a compact connected manifold $$\mathbb{M}$$ . We proved that if 0 < α < 1, α/2 < s < α and $$f \in {H^s}(\mathbb{M})$$ then the fractional Schrodinger propagator $${{\rm{e}}^{{\rm{i}}t{{\cal L}^{\alpha /2}}}}$$ on $$\mathbb{M}$$ satisfies $${{\rm{e}}^{{\rm{i}}t{{\cal L}^{\alpha /2}}}}f(x) - f(x) = o({t^{s/\alpha - \varepsilon }})$$ almost everywhere as t → 0+, for any e > 0.