Localization at every disorder strength in two dimensions
For $\eta > 0$ let $H_\eta$ be the Anderson model on $\ell^2(\mathbb{Z}^2)$, $$(H_\eta \psi)(n) = \sum_{|m - n|_1 = 1} \psi(m) + \eta V_n \psi(n),$$ where the $V_n$, $n \in \mathbb{Z}^2$, are independent and uniform on $[-1, 1]$. Almost surely its spectrum is $[-4 - \eta, 4 + \eta]$. The scaling theory of Abrahams, Anderson, Licciardello and Ramakrishnan predicts the absence of quantum diffusion in two dimensions. 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.
Problem. Is it true that for every $\eta > 0$, almost surely, $H_\eta$ has a complete orthonormal basis of eigenfunctions $\phi$ each satisfying $|\phi(n)| \le C_\phi e^{-c_\phi |n - n_\phi|}$ for all $n \in \mathbb{Z}^2$, for some constants $C_\phi, c_\phi > 0$ and some $n_\phi \in \mathbb{Z}^2$ depending on $\phi$? No decay rate uniform over the eigenfunctions is required.
Source: Simon (2000), Problem 2, as dense pure-point spectrum, after the physical prediction of Abrahams, Anderson, Licciardello and Ramakrishnan (1979). Stated by AIM — Open Applied Problems as exponential localization.
Imported from github.com/MColbrook/AIM.
Classification
- Primary
- SP — Spectral Theory
- Secondary
- MP, PR
- Keywords
- AIM Open Applied Problems, Anderson localization, 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{abrahams1979,
author = {Abrahams, E. and Anderson, P. W. and Licciardello, D. C. and Ramakrishnan, T. V.},
title = {Scaling theory of localization: absence of quantum diffusion in two dimensions},
journal = {Phys. Rev. Lett.},
volume = {42},
number = {10},
year = {1979},
pages = {673--676},
doi = {10.1103/PhysRevLett.42.673},
}
@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},
}
@misc{aim-anderson-planar-localization,
author = {Brunton, Steve and Colbrook, Matthew J. and de Hoop, Maarten V. and Stepaniants, George and Townsend, Alex and Ward, Rachel},
title = {{Localization at every disorder strength in two dimensions}},
howpublished = {AIM --- Open Applied Problems, GitHub repository},
year = {2026},
url = {https://github.com/MColbrook/AIM/blob/c9929805cece044705abaaebe1cebddac1ba11a6/problems/016-anderson-planar-localization.md},
note = {Entry 016 at commit c992980, 2026-10-05},
}
Citing this problem
This identifier is permanent. It will not change if the problem is solved, reclassified or withdrawn.
OP-00267, "Localization at every disorder strength in two dimensions", clopen.solutions.