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Zero Measure and Singular Continuous Spectra for Quantum Graphs

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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

  1. Non-compact, connected, and containing cycles.

  2. The diamond tiles can be viewed as circles of circumference 4. Similar graphs were recently considered in [4, 25].

  3. The scope of generality is comprehensively discussed in [21].

  4. In fact, “\(\subset \)” can be replaced by “\(=\)” by standard arguments utilizing ergodicity/minimality of \((\Omega ,T)\), cf., e.g., [14, 16].

  5. ± means that \(\tau \) is viewed as a differential expression on \((0,+\infty )\) and \((-\infty , 0)\) correspondingly.

  6. At this point \((\Omega , T)\) is not assumed to be a minimal subshift satisfying (B).

  7. Our operator is of the form described in [9, Section 3.2]; hence, the referenced result is applicable.

  8. i.e., \(e^{-cx}f(x)\in L^2({\mathbb {R}}_+, dx)\) for all \(c>0\)

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Author information

Authors and Affiliations

  1. Department of Mathematics, Rice University, Houston, TX, 77005, USA

    David Damanik, Licheng Fang & Selim Sukhtaiev

  2. Ocean University of China, Qingdao, 266100, Shandong, China

    Licheng Fang

  3. Rice University, Houston, TX, 77005, USA

    Licheng Fang

Authors
  1. David Damanik
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  2. Licheng Fang
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  3. Selim Sukhtaiev
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Correspondence to Selim Sukhtaiev.

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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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