Abstract
We study continuum Schrödinger operators on the real line whose potentials are comprised of two compactly supported square-integrable functions concatenated according to an element of the Fibonacci substitution subshift over two letters. We show that the Hausdorff dimension of the spectrum tends to one in the small coupling and high-energy regimes, regardless of the shape of the potential pieces.
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Acknowledgements
M. M. would like to express appreciation to Virginia Tech for hosting her for the Fall 2016 semester, when some of this work was conducted. The authors gratefully acknowledge Mark Embree for helpful conversations and the anonymous reviewers for helpful comments. Additionally, the authors would like to thank the Mathematical Research Institute at Oberwolfach for its hospitality during the Workshop: “Spectral Structures and Topological Methods in Mathematical Quasicrystals,” during which some of this work was completed.
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Communicated by Jean Bellissard.
J. F. was supported in part by an AMS-Simons travel grant, 2016–2018. M. M. was supported in part by Denison University and a Woodrow Wilson Career Enhancement Fellowship.
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Fillman, J., Mei, M. Spectral Properties of Continuum Fibonacci Schrödinger Operators. Ann. Henri Poincaré 19, 237–247 (2018). https://doi.org/10.1007/s00023-017-0624-8
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DOI: https://doi.org/10.1007/s00023-017-0624-8