Tilings without locally supported eigenfunctions in every MLD class
Let $\mathcal T$ be a tiling of the plane by polygons, with tile graph $\Gamma_{\mathcal T}$ (one vertex per tile, tiles adjacent when they share an edge) and graph Laplacian $\Delta_{\mathcal T} = D - A$. A locally supported eigenfunction is a finitely supported eigenfunction of $\Delta_{\mathcal T}$. As the source observes, it follows from work of Klassert, Lenz and Stollmann that every class of mutually locally derivable (MLD) tilings contains one whose Laplacian has a locally supported eigenfunction.
Problem. Can one always remove a locally supported eigenfunction from a Laplacian within its MLD class? Is it even true that every MLD class contains a tiling whose Laplacian has no locally supported eigenfunctions at all?
Source: Damanik, Embree, Fillman and Mei, Question 8.4 in “Discontinuities of the integrated density of states for Laplacians associated with Penrose and Ammann–Beenker tilings”, Experimental Mathematics 33 (2024); arXiv:2209.01443.
Imported from arxiv.org/abs/2209.01443.
Classification
- Primary
- SP — Spectral Theory
- Secondary
- DS, MP
- Keywords
- aperiodic order, graph Laplacian, locally supported eigenfunctions, mutual local derivability
- Literature checked
- Oct. 1, 2026 by clemens-admin
References
@article{damanik2024,
author = {Damanik, David and Embree, Mark and Fillman, Jake and Mei, May},
title = {Discontinuities of the integrated density of states for {L}aplacians associated with {P}enrose and {A}mmann--{B}eenker tilings},
journal = {Experimental Mathematics},
volume = {33},
number = {4},
pages = {588--610},
year = {2024},
doi = {10.1080/10586458.2023.2206589},
eprint = {2209.01443},
archivePrefix = {arXiv}
}
@article{klassert2003,
author = {Klassert, S. and Lenz, Daniel and Stollmann, Peter},
title = {Discontinuities of the integrated density of states for random operators on {D}elone sets},
journal = {Communications in Mathematical Physics},
volume = {241},
number = {2--3},
pages = {235--243},
year = {2003}
}
@book{baake2013,
author = {Baake, Michael and Grimm, Uwe},
title = {Aperiodic Order. {V}ol. 1: {A} Mathematical Invitation},
series = {Encyclopedia of Mathematics and its Applications},
volume = {149},
publisher = {Cambridge University Press},
year = {2013}
}
Citing this problem
This identifier is permanent. It will not change if the problem is solved, reclassified or withdrawn.
OP-00024, "Tilings without locally supported eigenfunctions in every MLD class", clopen.solutions.