Dispersive estimates for fractional order Schrödinger operators

We prove dispersive bounds for fractional Schrödinger operators on ${\mathbb{R}}^n$ of the form $H=(-\mathrm{\Delta }{)}^{\alpha }+V$ with $V$ a real-valued, decaying potential and $\alpha \mathrm{\notin }\mathbb{N}$. We derive pointwise bounds on the resolvent operators for all $0<\alpha <\frac{n}{2}$, a quantitative limiting absorption principle for $\frac{1}{2}<\alpha <\frac{n}{2}$, and establish global dispersive estimates in dimension $n\ge 2$ for the range $\frac{n+1}{4}\le \alpha <\frac{n}{2}$.