On the absolute continuity of Schrödinger operators with spherically symmetric, long-range potentials, I

Abstract

Let H = − Δ + V, where the potential V is spherically symmetric and can be decomposed as a sum of a short-range and a long-range term, V( r) = V S( r) + V L( r). Assume that for some r 0, V L( r) ϵ C 2 k ( r 0, ∞) and that there exist μ > 0, δ > 0, such that ( d dr ) jV L (r) = O(r −μ−jδ) as r → ∞, 1 ⩽ j ⩽ 2k . Assume further that min(2 kμ, (2 k − 1) δ + μ) > 1. Under this weak decay condition on V L( r) it is shown in this paper that the positive spectrum of H is absolutely continuous and that the absolutely continuous part of H is unitarily equivalent to −Δ, provided that the singularity of V at 0 is properly restricted. In particular, some oscillation of V L( r) at infinity is allowed.