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

Source abstract

We prove asymptotic completeness for operators of the form H=-Δ+L on L2(Rd), d≥2, where L is an admissible perturbation. Our class of admissible perturbations contains multiplication operators defined by real-valued potentials V∈Lq(Rd), 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