We prove asymptotic completeness for operators of the form $H=-\Delta+L$ on $L^2(\mathbb{R}^d)$ , $d\ge2$ , 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 $\vec{a}\cdot\nabla- \nabla\cdot\overline{\vec{a}}$ 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