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(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.
|Original language||English (US)|
|Number of pages||44|
|Journal||Duke Mathematical Journal|
|State||Published - Feb 15 2006|
All Science Journal Classification (ASJC) codes