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. https://resolver.caltech.edu/CaltechAUTHORS:20180315-130513176
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Abstract
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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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. | ||||||||||||
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Subject Keywords: | Neural Network; Statistical Physic; Complex System; Nonlinear Dynamics; Large Class | ||||||||||||
Issue or Number: | 1 | ||||||||||||
DOI: | 10.1007/BF02099468 | ||||||||||||
Record Number: | CaltechAUTHORS:20180315-130513176 | ||||||||||||
Persistent URL: | https://resolver.caltech.edu/CaltechAUTHORS:20180315-130513176 | ||||||||||||
Official Citation: | Hof, A., Knill, O. & Simon, B. Commun.Math. Phys. (1995) 174: 149. https://doi.org/10.1007/BF02099468 | ||||||||||||
Usage Policy: | No commercial reproduction, distribution, display or performance rights in this work are provided. | ||||||||||||
ID Code: | 85333 | ||||||||||||
Collection: | CaltechAUTHORS | ||||||||||||
Deposited By: | Ruth Sustaita | ||||||||||||
Deposited On: | 16 Mar 2018 14:42 | ||||||||||||
Last Modified: | 15 Nov 2021 20:27 |
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