Schrödinger operators defined by interval-exchange transformations

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April 2009, 3(2): 253-270. doi: 10.3934/jmd.2009.3.253

Schrödinger operators defined by interval-exchange transformations

Jon Chaika ^1, , David Damanik ^1, and Helge Krüger ^1,

1.	Department of Mathematics, Rice University, Houston, TX 77005, United States

Received September 2008 Published May 2009

Abstract
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We discuss discrete one-dimensional Schrödinger operators whose potentials are generated by an invertible ergodic transformation of a compact metric space and a continuous real-valued sampling function. We pay particular attention to the case where the transformation is a minimal interval-exchange transformation. Results about the spectral type of these operators are established. In particular, we provide the first examples of transformations for which the associated Schrödinger operators have purely singular spectrum for every nonconstant continuous sampling function.

Keywords: continuous spectrum, singular spectrum, interval-exchange transformations., Ergodic Schrödinger operators.

Mathematics Subject Classification: Primary 81Q10; Secondary 37A05, 47B36, 82B4.

Citation: Jon Chaika, David Damanik, Helge Krüger. Schrödinger operators defined by interval-exchange transformations. Journal of Modern Dynamics, 2009, 3 (2) : 253-270. doi: 10.3934/jmd.2009.3.253

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