Generic Continuous Spectrum for Ergodic Schrödinger Operators

  • Michael Boshernitzan
  • David DamanikEmail author


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.


Continuous Spectrum Spectral Type Arithmetic Progression Point Spectrum Sampling Function 
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Copyright information

© Springer-Verlag 2008

Authors and Affiliations

  1. 1.Department of MathematicsRice UniversityHoustonUSA

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