Singular continuous spectrum for palindromic Schrödinger operators
type: Item
instance of: scholarly article
title: Singular continuous spectrum for palindromic Schrödinger operators
zbMATH Open document ID: 0839.11009
DOI: 10.1007/BF02099468
author: O. Knill
author: B. Simon
author: A. L. Hof
published in: Communications in Mathematical Physics
publication date: 1996-06-19T00:00:00Z
review text: In this very interesting paper the authors study the discrete Schrödinger operator defined by \[ \bigl( L(x)u \bigr)_n = u_{n + 1} + u_{n - 1} + x_n u_n \quad \text{for } u \in \ell^2 (\mathbb{Z}), \] where \(x\) denotes an element of \(X\), a compact shift-invariant subset of \(A^\mathbb{Z}\), for \(A\) a finite set. This operator has been studied in particular when \(X\) is the closure of the orbit of a sequence generated by a finite automaton or by a morphism: for a general overview on the subject I would advise to read the paper by \textit{A. Sütö} [Schrödinger difference equation with deterministic ergodic potentials, in: Beyond quasicrystals, F. Axel (ed.) et al., Les Éditions de Physique, 481-549 (1995)]. The authors of the paper under review show that, if there exists a \(z\) in \(X\) that contains arbitrarily long palindromes, then there is a generic set in \(X\) for which the operator has purely singular continuous spectrum. The result holds in particular if \(X\) is the closure of the orbit of primitive substitutions in a large class (including Fibonacci, Thue-Morse, period doubling, binary and ternary non-Pisot). They address also the case of potentials defined by circle maps. Note that the results overlap results by \textit{A. Bovier} and \textit{J.-M. Ghez} [Commun. Math. Phys. 158, No. 1, 45-66 (1993; Zbl 0820.35099) and Erratum 166, No. 2, 431-432 (1994)] but no set of results is included in the other. The authors also conjecture that the Rudin-Shapiro sequence does not contain arbitrarily long palindromes and hence that their method cannot be applied for a Rudin-Shapiro sequence of potentials. This conjecture has been proved recently by the reviewer [Rapport de Recherche du LRI, No. 1035, 1996, Orsay].
Mathematics Subject Classification ID: 11B85
Mathematics Subject Classification ID: 35P05
Mathematics Subject Classification ID: 82B44
zbMATH DE Number: 821650
zbMATH Keywords: automatic sequences
zbMATH Keywords: sequences generated by morphism
zbMATH Keywords: discrete Schrödinger operator
zbMATH Keywords: singular continuous spectrum
zbMATH Keywords: Rudin-Shapiro sequence
reviewed by: Jean-Paul Allouche
MaRDI profile type: Publication
cites work: Q5751637
cites work: Spectral properties of a tight binding Hamiltonian with period doubling potential
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cites work: Spectral properties of one dimensional quasi-crystals
cites work: Spectral properties of one-dimensional Schrödinger operators with potentials generated by substitutions
cites work: Anderson localization for Bernoulli and other singular potentials
cites work: Symbolic dynamics for the renormalization map of a quasiperiodic Schrödinger equation
cites work: The spectrum of dynamical systems arising from substitutions of constant length
cites work: Absence of localization in a class of Schrödinger operators with quasiperiodic potential
cites work: Recurrence of the eigenstates of a Schrödinger operator with automatic potential
cites work: Some remarks on discrete aperiodic Schrödinger operators
cites work: 0-1-sequences of Toeplitz type
cites work: Operators with singular continuous spectrum. III: Almost periodic Schrödinger operators
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cites work: Sur les suites de Toeplitz
cites work: Substitution dynamical systems - spectral analysis
cites work: Operators with singular continuous spectrum. I: General operators
cites work: Schrödinger semigroups
cites work: The spectrum of a quasiperiodic Schrödinger operator
cites work: Singular continuous spectrum on a Cantor set of zero Lebesgue measure for the Fibonacci Hamiltonian
full work available at URL: https://doi.org/10.1007/bf02099468
OpenAlex ID: W2169978469
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description: scientific article; zbMATH DE number 821650