The $L^p$-continuity of wave operators for fractional order Schrödinger operators

Abstract

We consider fractional Schrödinger operators $H=(-\Delta {)}^{\alpha }+V(x)$ in $n$ dimensions with real-valued potential $V$ when $n>2\alpha$, $\alpha >1$. We show that the wave operators extend to bounded operators on $L^p({\mathbb{R}}^n)$ for all $1\le p\le \infty$ under conditions on the potential that depend on $n$ and $\alpha$ analogously to the case when $\alpha \in \mathbb{N}$. As a consequence, we deduce a family of dispersive and Strichartz estimates for the perturbed fractional Schrödinger operator.

Updated to correct typos and reflect referee comments. 30 pages, submitted