Agmon-Kato-Kuroda theorems for a large class of perturbations

Abstract

We prove asymptotic completeness for operators of the form $H=-\Delta +L$ on $L^2({\mathbb{R}}^d)$, $d\ge 2$, where $L$ is an admissible perturbation. Our class of admissible perturbations contains multiplication operators defined by real-valued potentials $V\in L^q({\mathbb{R}}^d)$, $q\in [d/2,(d+1)/2]$ (if $d=2$, then we require $q\in (1,3/2]$), as well as real-valued potentials $V$ satisfying a global Kato condition. The class of admissible perturbations also contains first-order differential operators of the form $\overrightarrow{a}·\nabla -\nabla ·\stackrel{̲}{\overrightarrow{a}}$ for suitable vector potentials $a$. Our main technical statement is a new limiting absorption principle, which we prove using techniques from harmonic analysis related to the Stein-Tomas restriction theorem