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Uniform spectral properties of one-dimensional quasicrystals, iv. quasi-sturmian potentials

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Abstract

We consider discrete one-dimensional Schrödinger operators with quasi-Sturmian potentials. We present a new approach to the trace map dynamical system which is independent of the initial conditions and establish a characterization of the spectrum in terms of bounded trace map orbits. Using this, it is shown that the operators have purely singular continuous spectrum and their spectrum is a Cantor set of Lebesgue measure zero. We also exhibit a subclass having purely α-continuous spectrum. All these results hold uniformly on the hull generated by a given potential.

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Authors and Affiliations

  1. Department of Mathematics, University of California, 92697, Irvine, CA, USA

    David Damanik

  2. Institute of Mathematics, The Hebrew University of Jerusalem, 91904, Jerusalem, Israel

    Daniel Lenz

  3. Fachbereich Mathematik, Johann Wolfgang Goethe-UniversitÄt, 60054, Frankfurt, Germany

    Daniel Lenz

Authors
  1. David Damanik
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  2. Daniel Lenz
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Additional information

Research of D. L. was supported in part by The Israel Science Foundation (grant no. 447/99) and the Edmund Landau Center for Research in Mathematical Analysis and Related Areas, sponsored by the Minerva Foundation (Germany).

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Damanik, D., Lenz, D. Uniform spectral properties of one-dimensional quasicrystals, iv. quasi-sturmian potentials. J. Anal. Math. 90, 115–139 (2003). https://doi.org/10.1007/BF02786553

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  • DOI: https://doi.org/10.1007/BF02786553

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