Localization and unique continuation for the Anderson-Bernoulli model with long-range hopping on $\mathbb{Z}$

Shihe Liu; Yunfeng Shi; Zhifei Zhang

Localization and unique continuation for the Anderson-Bernoulli model with long-range hopping on $\mathbb{Z}$

Spectral Theory 2026-07-03 v1 Mathematical Physics Analysis of PDEs Dynamical Systems Probability

Authors: Shihe Liu , Yunfeng Shi , Zhifei Zhang

Abstract

In this paper, we study Anderson localization near the spectral edge for the Anderson-Bernoulli model on $\mathbb{Z}$ with long-range hopping. When the hopping has a rational Laurent symbol, a quantitative version of the unique continuation principle can be proved, and localization occurs. For the unique continuation in the general case, we give some counterexamples and prove a weaker result for hopping that decays faster than exponential rate. To the best of our knowledge, this is the first localization result for the long-range Anderson model with pure Bernoulli potentials.

Keywords

Anderson localization Anderson-Bernoulli model long-range hopping unique continuation principle spectral edge Bernoulli potentials

Cite

@article{arxiv.2607.03472,
  title  = {Localization and unique continuation for the Anderson-Bernoulli model with long-range hopping on $\mathbb{Z}$},
  author = {Shihe Liu and Yunfeng Shi and Zhifei Zhang},
  journal= {arXiv preprint arXiv:2607.03472},
  year   = {2026}
}