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

Matania Ben-Artzi*

*Corresponding author for this work

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14 Scopus citations

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) = VS(r) + VL(r). Assume that for some r0, VL(r) ϵ C2k(r0, ∞) and that there exist μ 〉 0, δ 〉 0, such that (d/dr)jVL(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 VL(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 VL(r) at infinity is allowed.

Original languageEnglish
Pages (from-to)41-50
Number of pages10
JournalJournal of Differential Equations
Volume38
Issue number1
DOIs
StatePublished - 1980
Externally publishedYes

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