A nonprimitive substitution Schrödinger operator with generic singular continuous spectrum
type: Item
instance of: scholarly article
title: A nonprimitive substitution Schrödinger operator with generic singular continuous spectrum
zbMATH Open document ID: 1002.11028
DOI: 10.1016/S0034-4877(00)80009-9
author: César R. de Oliveira
author: Marcus V. Lima
published in: Reports on Mathematical Physics
publication date: 2001-02-08T00:00:00Z
review text: The nature of the spectrum of a discrete one-dimensional Schrödinger operator with potentials generated by substitutions has been studied by several authors. We quote in particular the paper by \textit{A. Hof, O. Knill} and \textit{B. Simon}, where they prove that the spectrum is generically purely singular continuous when the sequence of potentials is generated by a primitive morphism (substitution), and contains arbitrarily long palindromes [Commun. Math. Phys. 174, 149-159 (1995; Zbl 0839.11009)]. Here the authors study the morphism \(a\to aabaa\), \(b\to b\) that is not primitive. They prove nevertheless that the spectrum of the corresponding Schrödinger operator is generically purely singular continuous: the main ingredient (in comparison with the paper of Hof, Knill and Simon) is to prove that the discrete dynamical system associated with \(a\to aabaa\), \(b\to b\) is minimal. Note that it would be interesting to try to obtain the result by using the work of \textit{A. Bovier} and \textit{J.-M. Ghez} [see Commun. Math. Phys. 158, 45-66 (1993; Zbl 0820.35099) and Errata in ibid. 166, 431-432 (1994; Zbl 0841.35072)].
Mathematics Subject Classification ID: 11B85
Mathematics Subject Classification ID: 34L40
Mathematics Subject Classification ID: 82B44
Mathematics Subject Classification ID: 47B39
Mathematics Subject Classification ID: 68R15
zbMATH DE Number: 1507663
zbMATH Keywords: substitutive sequences
zbMATH Keywords: automata sequences
zbMATH Keywords: spectrum
zbMATH Keywords: discrete one-dimensional Schrödinger operator
reviewed by: Jean-Paul Allouche
MaRDI profile type: Publication
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full work available at URL: s0034-4877(00)80009-9
OpenAlex ID: W2036756374
description: scientific article; zbMATH DE number 1507663