Extended states in the three-dimensional Anderson model
For $\eta > 0$ let $H_\eta$ be the Anderson model on $\ell^2(\mathbb{Z}^3)$, $$(H_\eta \psi)(n) = \sum_{|m - n|_1 = 1} \psi(m) + \eta V_n \psi(n),$$ where the $V_n$, $n \in \mathbb{Z}^3$, are independent and uniform on $[-1, 1]$. Almost surely its spectrum is $[-6 - \eta, 6 + \eta]$. Fröhlich, Martinelli, Scoppola and Spencer, and later Aizenman and Molchanov, proved that for large enough $\eta$ the spectrum is almost surely pure point with exponentially decaying eigenfunctions. On the Bethe lattice, Klein proved that at weak disorder the analogous model has purely absolutely continuous spectrum in an energy interval.
Problem. Do there exist $\eta_0 > 0$ and a nonempty open interval $I \subset (-6, 6)$ such that, for every $0 < \eta < \eta_0$, almost surely the restriction of $H_\eta$ to its spectral subspace for $I$ is purely absolutely continuous and $I \subset \sigma(H_\eta)$?
Source: Simon (2000), Problem 1 (extended states), for dimension at least three and suitable disorder widths. Stated by AIM — Open Applied Problems in dimension three, for small symmetric uniform disorder and a fixed energy interval.
Imported from github.com/MColbrook/AIM.
Classification
- Primary
- SP — Spectral Theory
- Secondary
- MP, PR
- Keywords
- absolutely continuous spectrum, AIM Open Applied Problems, Anderson model, random Schrödinger operators
References
@incollection{simon2000,
author = {Simon, Barry},
title = {Schr{\"o}dinger operators in the twenty-first century},
booktitle = {Mathematical Physics 2000},
editor = {Fokas, Athanassios and Grigoryan, Alexander and Kibble, Tom and Zegarlinski, Boguslaw},
publisher = {Imperial College Press},
address = {London},
year = {2000},
pages = {283--288},
doi = {10.1142/9781848160224_0014},
}
@article{frohlich1985,
author = {Fr{\"o}hlich, J. and Martinelli, F. and Scoppola, E. and Spencer, T.},
title = {Constructive proof of localization in the {A}nderson tight binding model},
journal = {Comm. Math. Phys.},
volume = {101},
number = {1},
year = {1985},
pages = {21--46},
doi = {10.1007/BF01212355},
}
@article{aizenman1993,
author = {Aizenman, Michael and Molchanov, Stanislav},
title = {Localization at large disorder and at extreme energies: an elementary derivation},
journal = {Comm. Math. Phys.},
volume = {157},
number = {2},
year = {1993},
pages = {245--278},
doi = {10.1007/BF02099760},
}
@article{klein1998,
author = {Klein, Abel},
title = {Extended states in the {A}nderson model on the {B}ethe lattice},
journal = {Adv. Math.},
volume = {133},
number = {1},
year = {1998},
pages = {163--184},
doi = {10.1006/aima.1997.1688},
}
@misc{aim-anderson-extended-states,
author = {Brunton, Steve and Colbrook, Matthew J. and de Hoop, Maarten V. and Stepaniants, George and Townsend, Alex and Ward, Rachel},
title = {{Extended states in the three-dimensional Anderson model}},
howpublished = {AIM --- Open Applied Problems, GitHub repository},
year = {2026},
url = {https://github.com/MColbrook/AIM/blob/1b7c7fd251c1643bb41f0e3cef38704bcef410fd/problems/015-anderson-extended-states.md},
note = {Entry 015 at commit 1b7c7fd, 2026-10-13},
}
Resolution claims
For each fixed dimension $d\ge3$ and each fixed sufficiently small positive disorder strength, we prove that the Anderson operator on $\mathbb Z^d$ with independent uniform site potentials almost surely has purely absolutely continuous spectrum with nonzero weight on an open energy interval. The interval may depend on $d$ but is independent of the disorder strength. This settles the purely absolutely …
Status history
| When | Change | By | Reason |
|---|---|---|---|
| 2026-10-16 | Needs verification → Solution claimed | — | Recorded here: math @ 1581d4c, build/paper.tex. |
Citing this problem
This identifier is permanent. It will not change if the problem is solved, reclassified or withdrawn.
OP-00663, "Extended states in the three-dimensional Anderson model", clopen.solutions.