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

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. Let λ = lim sup_r→∞ V_L(r) < ∞ (we allow λ = − ∞) and set λ⁺ = max(λ, 0). Assume that for some r₀, V_L(r) ϵ C^2k(r₀, ∞) and that there exists δ 〉 0 such that (d/dr)^jV_L(r) · (λ⁺ − V_L(r) + 1)⁻¹ = O(r^−jδ), j = 1,…, 2k, as r → ∞. Assume further that ∝₁^∞(dr/¦ V_L(r)¦^1/2) = ∞ and that 2kδ 〉 1. It is shown that: (a) The restriction of H to C^∞(Rⁿ) is essentially self-adjoint, (b) The essential spectrum of H contains the closure of (λ, ∞). (c) The part of H over (λ, ∞) is absolutely continuous.