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

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 r₀, V_L(r) ϵ C^2k(r₀, ∞) 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(2kμ, (2k − 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.