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All-Disorder Localization for the One-Dimensional Anderson Model with Power-Law Hopping
Abstract
We consider the one-dimensional random operator (Hλ,ωu)(n) = ∑m≠ nu(m)/(|n−m|r) +λωnu(n), where the random variables ωn are independent and uniformly distributed on [−1,1] (i.e., \mbox{ωnstackreli.i.d.∼Unif[−1,1]}). For every λ>0 and r>3/2, we prove that Hλ,ω almost surely has pure point spectrum and admits an orthonormal eigenbasis {ψj} satisfying ∑n∈bbZ(1+|n|)2τ|ψj(n)|2<∞ for every j and 0≤τ<r−3/2. These conclusions establish spectral localization, thereby rigorously confirming the prediction of earlier analytical and numerical studies in the physics literature, which identified r=3/2 as the marginal exponent.
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BibTeX
@misc{jiang2026alldisorder,
title = {{All-Disorder Localization for the One-Dimensional Anderson Model with Power-Law Hopping}},
author = {Huajian Jiang and Yunfeng Shi},
howpublished = {arXiv},
year = {2026},
month = nov,
doi = {10.48550/arxiv.2611.02724},
eprint = {2611.02724},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2611.02724},
}
RIS
TY - UNPB
TI - All-Disorder Localization for the One-Dimensional Anderson Model with Power-Law Hopping
AU - Huajian Jiang
AU - Yunfeng Shi
PY - 2026
DA - 2026/11/02
PB - arXiv
DO - 10.48550/arxiv.2611.02724
UR - https://arxiv.org/abs/2611.02724
AB - We consider the one-dimensional random operator (H_λ,ωu)(n) = ∑_m≠ nu(m)/(|n−m|^r) +λω_nu(n), where the random variables ω_n are independent and uniformly distributed on [−1,1] (i.e., \mbox{ω_nstackreli.i.d.∼Unif[−1,1]}). For every λ>0 and r>3/2, we prove that H_λ,ω almost surely has pure point spectrum and admits an orthonormal eigenbasis {ψ_j} satisfying ∑_n∈bbZ(1+|n|)^2τ|ψ_j(n)|^2<∞ for every j and 0≤τ<r−3/2. These conclusions establish spectral localization, thereby rigorously confirming the prediction of earlier analytical and numerical studies in the physics literature, which identified r=3/2 as the marginal exponent.
ER -