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Singular Continuous Spectrum for Palindromic Schrödinger Operators

Hof, A. and Knill, O. and Simon, B. (1995) Singular Continuous Spectrum for Palindromic Schrödinger Operators. Communications in Mathematical Physics, 174 (1). pp. 149-159. ISSN 0010-3616. doi:10.1007/BF02099468.

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We give new examples of discrete Schrödinger operators with potentials taking finitely many values that have purely singular continuous spectrum. If the hull X of the potential is strictly ergodic, then the existence of just one potentialx in X for which the operator has no eigenvalues implies that there is a generic set in X for which the operator has purely singular continuous spectrum. A sufficient condition for the existence of such anx is that there is a z ∈ 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 in X. 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 all x ∈ X if X derives from a primitive substitution. For potentials defined by circle maps, x_n = 1_J (θ_0 + nα), we show that the operator has purely singular continuous spectrum for a generic subset in X for all irrational α and every half-open interval J.

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Simon, B.0000-0003-2561-8539
Additional Information:© 1995 Springer-Verlag. Received: 28 July 1994; in revised form: 28 November 1994. 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-9101715. The Government has certain rights in this material.
Funding AgencyGrant Number
Natural Sciences and Engineering Research Council of Canada (NSERC)UNSPECIFIED
Subject Keywords:Neural Network; Statistical Physic; Complex System; Nonlinear Dynamics; Large Class
Issue or Number:1
Record Number:CaltechAUTHORS:20180315-130513176
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Official Citation:Hof, A., Knill, O. & Simon, B. Commun.Math. Phys. (1995) 174: 149.
Usage Policy:No commercial reproduction, distribution, display or performance rights in this work are provided.
ID Code:85333
Deposited By: Ruth Sustaita
Deposited On:16 Mar 2018 14:42
Last Modified:15 Nov 2021 20:27

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