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 az∈X 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 allx∈X ifX derives from a primitive substitution. For potentials defined by circle maps,x n =1 J (θ0+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.
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Communicated by A. Jaffe
Work partially supported by NSERC.
This material is based upon work supported by the National Science Foundation under Grant No. DMS-91-1715. The Government has certain rights in this material.
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Hof, A., Knill, O. & Simon, B. Singular continuous spectrum for palindromic Schrödinger operators. Commun.Math. Phys. 174, 149–159 (1995). https://doi.org/10.1007/BF02099468
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DOI: https://doi.org/10.1007/BF02099468