Dispersive estimates for fractional order Schrödinger operators

Abstract

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

Expanded introduction. To appear in Communications of the AMS. 33 pages