Agmon-kato-kuroda theorems for a large class of perturbations

We prove asymptotic completeness for operators of the form H = -Δ + L on L²(ℝ^d), d ≥ 2, where L is an admissible perturbation. Our class of admissible perturbations contains multiplication operators defined by real-valued potentials V ∈ L^q(ℝ ^d), q ∈ [d/2, (d + 1)/2] (if d = 2, then we require q ∈ (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 a · ∇ - ∇ · 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.