Spectral properties of aperiodic metric and discrete graphs

[2608.27725] Spectral properties of aperiodic metric and discrete graphs Skip to main content Search Submit Donate Log in Mathematical Physics arXiv:2608.27725 (math-ph) [Submitted on 27 Aug 2026] Title: Spectral properties of aperiodic metric and discrete graphs Authors: Gilad Sofer View PDF HTML (experimental) Abstract: In this thesis, we study the spectral properties of dynamically defined aperiodic metric and discrete graphs. Our goal is to determine to what extent spectral properties of discrete one-dimensional ergodic Schrödinger operators persist when the aperiodicity is manifested through the geometry rather than through a potential. The graphs considered here are inspired by one-dimensional aperiodic tilings, and are called tiling graphs and decorated $\mathbb{Z}$-graphs. For a large family of metric tiling graphs equipped with the standard Laplacian, we show that the spectrum is of zero Lebesgue measure, and is a generalized Cantor set up to a possible discrete set of energies. For decorated $\mathbb{Z}$-graphs, we further show that for a Baire-generic and Lebesgue almost-sure choice of the decoration edge lengths, the spectrum is a generalized Cantor set. We then study the integrated density of states (IDS) for metric and discrete decorated $\mathbb{Z}$-graphs. We prove a gap labelling theorem, which characterizes the set of possible values taken by the IDS inside spectral gaps. We show that the gap labels are contained in the Schwartzman group associated with the dynamical system generating the graph, up to a geometric scaling factor. Lastly, we consider the Dry Ten Martini Problem for discrete Sturmian decorated $\mathbb{Z}$-graphs, asking whether all possible values predicted by the gap labelling theorem are indeed attained by the IDS inside spectral gaps. We answer this question negatively, by identifying a large set of gap labels which are not attained due to jump discontinuities of the IDS. We then show that away from these jump discontinuities, the periodic approximants for Sturmian graphs display the same combinatorial structure as the standard Sturmian Hamiltonians, and use this to obtain an explicit characterization of the realized gap labels for Sturmian comb graphs. Subjects: Mathematical Physics (math-ph) ; Spectral Theory (math.SP) Cite as: arXiv:2608.27725 [math-ph] (or arXiv:2608.27725v1 [math-ph] for this version) https://doi.org/10.48550/arXiv.2608.27725 Focus to learn more arXiv-issued DOI via DataCite Submission history From: Gilad Sofer [ view email ] [v1] Thu, 27 Aug 2026 21:32:41 UTC (3,889 KB) Full-text links: Access Paper: View PDF HTML (experimental) TeX Source view license Current browse context: math-ph < prev | next > new | recent | 2026-08 Change to browse by: math math.MP math.SP References & Citations SciX Google Scholar Semantic Scholar export BibTeX citation Bookmark Bibliographic Tools Bibliographic and Citation Tools Bibliographic Explorer Toggle Bibliographic Explorer ( What is the Explorer? ) Connected Papers Toggle Connected Papers ( What is Connected Papers? ) Litmaps Toggle Litmaps ( What is Litmaps? ) scite.ai Toggle scite Smart Citations ( What are Smart Citations? ) Code, Data, Media Code, Data and Media Associated with this Article alphaXiv Toggle alphaXiv ( What is alphaXiv? ) Links to Code Toggle CatalyzeX Code Finder for Papers ( What is CatalyzeX? ) DagsHub Toggle DagsHub ( What is DagsHub? ) GotitPub Toggle Gotit.pub ( What is GotitPub? ) Huggingface Toggle Hugging Face ( What is Huggingface? ) ScienceCast Toggle ScienceCast ( What is ScienceCast? ) Demos Demos Replicate Toggle Replicate ( What is Replicate? ) Spaces Toggle Hugging Face Spaces ( What is Spaces? ) Spaces Toggle TXYZ.AI ( What is TXYZ.AI? ) Related Papers Recommenders and Search Tools Link to Influence Flower Influence Flower ( What are Influence Flowers? ) Core recommender toggle CORE Recommender ( What is CORE? ) Author Venue Institution Topic About arXivLabs arXivLabs: experimental projects with community collaborators arXivLabs is a framework that allows collaborators to develop and share new arXiv features directly on our website. Both individuals and organizations that work with arXivLabs have embraced and accepted our values of openness, community, excellence, and user data privacy. arXiv is committed to these values and only works with partners that adhere to them. Have an idea for a project that will add value for arXiv's community? Learn more about arXivLabs . Which authors of this paper are endorsers? | Disable MathJax ( What is MathJax? ) We gratefully acknowledge support from our major funders , member institutions , and all contributors. About · Help · Contact · Subscribe · Copyright · Privacy · Accessibility · Operational Status (opens in new tab)

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