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

Authors
Journal
Journal of Differential Equations
0022-0396
Publisher
Elsevier
Publication Date
Volume
38
Issue
1
Identifiers
DOI: 10.1016/0022-0396(80)90023-6

Abstract

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.

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