Anderson localization for the cosine skew-shift model at every coupling
Let $\alpha \in \mathbb{R}$ be Diophantine: there are $c, \tau > 0$ such that $\|q\alpha\|_{\mathbb{R}/\mathbb{Z}} \ge c|q|^{-\tau}$ for every nonzero integer $q$. For $\lambda > 0$ and $(x, y) \in (\mathbb{R}/\mathbb{Z})^2$ let $H_{x,y}$ act on $\ell^2(\mathbb{Z})$ by $$(H_{x,y}\psi)_n = \psi_{n+1} + \psi_{n-1} + 2\lambda \cos\!\left(2\pi\left[y + nx + \tfrac{n(n-1)}{2}\alpha\right]\right)\psi_n,$$ the potential generated by the skew-shift $(x, y) \mapsto (x + \alpha, x + y)$ of the 2-torus. Bourgain, Goldstein and Schlag proved Anderson localization at large coupling, for frequencies outside a set of small measure, when the cosine is replaced by any nonconstant real-analytic function on the 2-torus. Bourgain showed that there is an explicit phase $(x, y)$ such that for every $\lambda > 0$ there is a set of $\alpha$ of positive measure for which this operator has eigenvalues whose closure has positive Lebesgue measure.
Problem. Is it true that for every such $\alpha$, every $\lambda > 0$ and Lebesgue-almost every $(x, y)$, $H_{x,y}$ has an orthonormal basis of eigenvectors $\psi$, each satisfying $|\psi_n| \le C_\psi e^{-c_\psi |n|}$ for all $n \in \mathbb{Z}$, for some constants $C_\psi, c_\psi > 0$?
Source: Damanik (2017), Section 9.1, the Skew-Shift-Problem, of which spectral localization is one part; stated for Diophantine frequencies by AIM — Open Applied Problems.
Imported from github.com/MColbrook/AIM.
Classification
- Primary
- SP — Spectral Theory
- Secondary
- DS, MP
- Keywords
- AIM Open Applied Problems, Anderson localization, discrete Schrödinger operators, skew-shifts
References
@article{damanik2017,
author = {Damanik, David},
title = {Schr{\"o}dinger operators with dynamically defined potentials},
journal = {Ergodic Theory Dynam. Systems},
volume = {37},
number = {6},
year = {2017},
pages = {1681--1764},
doi = {10.1017/etds.2015.120},
eprint = {1410.2445},
archivePrefix = {arXiv},
}
@article{bourgain2001,
author = {Bourgain, Jean and Goldstein, Michael and Schlag, Wilhelm},
title = {Anderson localization for {S}chr{\"o}dinger operators on {$\mathbb{Z}$} with potentials given by the skew-shift},
journal = {Comm. Math. Phys.},
volume = {220},
number = {3},
year = {2001},
pages = {583--621},
doi = {10.1007/PL00005570},
}
@article{bourgain2002,
author = {Bourgain, Jean},
title = {On the spectrum of lattice {S}chr{\"o}dinger operators with deterministic potential},
journal = {J. Anal. Math.},
volume = {87},
year = {2002},
pages = {37--75},
doi = {10.1007/BF02868469},
}
@misc{aim-skew-shift-localization,
author = {Brunton, Steve and Colbrook, Matthew J. and de Hoop, Maarten V. and Stepaniants, George and Townsend, Alex and Ward, Rachel},
title = {{Anderson localization for the cosine skew-shift model at every coupling}},
howpublished = {AIM --- Open Applied Problems, GitHub repository},
year = {2026},
url = {https://github.com/MColbrook/AIM/blob/a153f469ee0ea64e876b2ad97c79d252c19a3e22/problems/113-skew-shift-localization.md},
note = {Entry 113 at commit a153f46, 2026-10-13},
}
Citing this problem
This identifier is permanent. It will not change if the problem is solved, reclassified or withdrawn.
OP-00383, "Anderson localization for the cosine skew-shift model at every coupling", clopen.solutions.