Agmon–kato–kuroda Theorems for a Large Class of Perturbations

We prove asymptotic completeness for operators of the form H = −∆+L on L(R), d ≥ 2, where L is an admissible perturbation. Our class of admissible perturbations contains multiplication operators defined by real-valued potentials V ∈ L(R), 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.