OpenAlex ·
Zero measure Cantor spectrum for aperiodic metric tiling graphs
- Published
- Source
- OpenAlex
- Published in
- arXiv (Cornell University)
- DOI
- 10.48550/arxiv.2610.02650
- Original
- Read it at the publisher
- Topics
Abstract
We study a family of aperiodic metric graphs generated by one-dimensional tilings. We prove that, generically, the spectrum of the associated Kirchhoff Laplacian has Lebesgue measure zero and, apart from a discrete set of isolated eigenvalues, is a generalized Cantor set. For a large subclass of these graphs, we further show that isolated eigenvalues are generically absent. This extends several well-known results from the setting of discrete ergodic Schrödinger operators on $\mathbb{Z}$ to a large family of metric graphs. Earlier results of this kind for metric graphs typically assume equilateral edges and exploit symmetries that reduce the analysis to a Sturm-Liouville operator on the real line. By contrast, the graph families considered here need not have equilateral edges or the symmetries needed for such a reduction and therefore require a different analytical approach.
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