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Generic Continuous Spectrum for Ergodic Schrödinger Operators

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Abstract

We consider families of discrete Schrödinger operators on the line with potentials generated by a homeomorphism on a compact metric space and a continuous sampling function. We introduce the concepts of topological and metric repetition property. Assuming that the underlying dynamical system satisfies one of these repetition properties, we show using Gordon’s Lemma that for a generic continuous sampling function, the set of elements in the associated family of Schrödinger operators that have no eigenvalues is large in a topological or metric sense, respectively. We present a number of applications, particularly to shifts and skew-shifts on the torus.

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Authors and Affiliations

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

    Michael Boshernitzan & David Damanik

Authors
  1. Michael Boshernitzan
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  2. David Damanik
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Correspondence to David Damanik.

Additional information

Communicated by B. Simon

D. D. was supported in part by NSF grant DMS–0653720.

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Boshernitzan, M., Damanik, D. Generic Continuous Spectrum for Ergodic Schrödinger Operators. Commun. Math. Phys. 283, 647–662 (2008). https://doi.org/10.1007/s00220-008-0537-y

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