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Journal of Modern Dynamics (JMD)
 

Schrödinger operators defined by interval-exchange transformations

Pages: 253 - 270, Issue 2, April 2009      doi:10.3934/jmd.2009.3.253

 
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Jon Chaika - Department of Mathematics, Rice University, Houston, TX 77005, United States (email)
David Damanik - Department of Mathematics, Rice University, Houston, TX 77005, United States (email)
Helge Krüger - Department of Mathematics, Rice University, Houston, TX 77005, United States (email)

Abstract: 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:  Ergodic Schrödinger operators, singular spectrum, continuous spectrum, interval-exchange transformations.
Mathematics Subject Classification:  Primary 81Q10; Secondary 37A05, 47B36, 82B44.

Received: September 2008;      Available Online: May 2009.