TY - JOUR
T1 - On the absolute continuity of Schrödinger operators with spherically symmetric, long-range potentials, I
AU - Ben-Artzi, Matania
PY - 1980
Y1 - 1980
N2 - 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.
AB - 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.
UR - http://www.scopus.com/inward/record.url?scp=0001946271&partnerID=8YFLogxK
U2 - 10.1016/0022-0396(80)90023-6
DO - 10.1016/0022-0396(80)90023-6
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AN - SCOPUS:0001946271
SN - 0022-0396
VL - 38
SP - 41
EP - 50
JO - Journal of Differential Equations
JF - Journal of Differential Equations
IS - 1
ER -