Abstract
We introduce a dynamically defined class of unbounded, connected, equilateral metric graphs on which the Kirchhoff Laplacian has zero Lebesgue measure spectrum and a non-trivial singular continuous part. A local Borg–Marchenko uniqueness result is obtained in order to utilize Kotani theory for aperiodic subshifts satisfying Boshernitzan’s condition.
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Notes
Non-compact, connected, and containing cycles.
The scope of generality is comprehensively discussed in [21].
± means that \(\tau \) is viewed as a differential expression on \((0,+\infty )\) and \((-\infty , 0)\) correspondingly.
At this point \((\Omega , T)\) is not assumed to be a minimal subshift satisfying (B).
Our operator is of the form described in [9, Section 3.2]; hence, the referenced result is applicable.
i.e., \(e^{-cx}f(x)\in L^2({\mathbb {R}}_+, dx)\) for all \(c>0\)
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Communicated by Jan Derezinski.
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D. Damanik was supported in part by NSF Grant DMS–1700131. L. Fang was supported by NSFC (No. 11571327) and the Joint Ph.D. Scholarship Program of Ocean University of China. S. Sukhtaiev was supported in part by an AMS-Simons travel Grant, 2017–2019.
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Damanik, D., Fang, L. & Sukhtaiev, S. Zero Measure and Singular Continuous Spectra for Quantum Graphs. Ann. Henri Poincaré 21, 2167–2191 (2020). https://doi.org/10.1007/s00023-020-00920-6
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DOI: https://doi.org/10.1007/s00023-020-00920-6