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Communications in Mathematical Physics

, Volume 174, Issue 1, pp 149–159 | Cite as

Singular continuous spectrum for palindromic Schrödinger operators

  • A. Hof
  • O. Knill
  • B. Simon
Article

Abstract

We give new examples of discrete Schrödinger operators with potentials taking finitely many values that have purely singular continuous spectrum. If the hullX of the potential is strictly ergodic, then the existence of just one potentialx inX for which the operator has no eigenvalues implies that there is a generic set inX for which the operator has purely singular continuous spectrum. A sufficient condition for the existence of such anx is that there is azX that contains arbitrarily long palindromes. Thus we can define a large class of primitive substitutions for which the operators are purely singularly continuous for a generic subset inX. The class includes well-known substitutions like Fibonacci, Thue-Morse, Period Doubling, binary non-Pisot and ternary non-Pisot. We also show that the operator has no absolutely continuous spectrum for allxX ifX derives from a primitive substitution. For potentials defined by circle maps,x n =1 J 0+nα), we show that the operator has purely singular continuous spectrum for a generic subset inX for all irrational α and every half-open intervalJ.

Keywords

Neural Network Statistical Physic Complex System Nonlinear Dynamics Large Class 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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Copyright information

© Springer-Verlag 1995

Authors and Affiliations

  • A. Hof
    • 1
  • O. Knill
    • 2
  • B. Simon
    • 2
  1. 1.Department of Mathematics and StatisticsMcMaster UniversityHamiltonCanada
  2. 2.Division of Physics, Mathematics and AstronomyCalifornia Institute of TechnologyPasadenaUSA

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