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

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. Let λ = lim sup r→∞ V L( r) < ∞ (we allow λ = − ∞) and set λ + = max( λ, 0). Assume that for some r 0, V L( r) ϵ C 2 k ( r 0, ∞) 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 2 kδ > 1. It is shown that: (a) The restriction of H to C ∞( R n ) 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.