Spectral properties of one-dimensional Schrödinger operators with potentials generated by substitutions
type: Item
instance of: scholarly article
title: Spectral properties of one-dimensional Schrödinger operators with potentials generated by substitutions
zbMATH Open document ID: 0820.35099
DOI: 10.1007/BF02097231
author: Jean-Michel Ghez
author: Anton Bovier
published in: Communications in Mathematical Physics
publication date: 1995-09-21T00:00:00Z
review text: In this very interesting paper the authors study the spectra of one- dimensional discrete Schrödinger operators whose potentials are fixed point of substitutions. The usual method is to study the dynamical system generated by the trace map: this map can be defined in terms of the substitution giving the sequence of potentials, but it is often rather complicated to study. The nice idea of the authors is to introduce a so-called ``reduced trace map'' and to associate in a natural way to it a (new) substitution. If this new substitution is ``semi-primitive'' (a notion proposed by the authors which generalizes the notion of primitivity), if the initial substitution is primitive, if furthermore the sequence of potentials contains a square (i.e. two identical consecutive blocks), then the main result of the authors asserts that the spectrum of the associated Schrödinger operator is singular and supported on a set of zero Lebesgue measure. They also give a condition to exclude point spectrum. Finally they give many examples (known or not) where their result can be applied. They show at the end of the paper that the Rudin-Shapiro sequence ``retains its mystery'' as it does not enter this framework.
Mathematics Subject Classification ID: 35P05
Mathematics Subject Classification ID: 35J10
Mathematics Subject Classification ID: 82B44
Mathematics Subject Classification ID: 47F05
Mathematics Subject Classification ID: 11B85
zbMATH DE Number: 490659
zbMATH Keywords: substitution
zbMATH Keywords: trace map
zbMATH Keywords: spectra of one-dimensional discrete Schrödinger operators
reviewed by: Jean-Paul Allouche
MaRDI profile type: Publication
cites work: Substitution dynamical systems - spectral analysis
cites work: Symbolic dynamics for the renormalization map of a quasiperiodic Schrödinger equation
cites work: The spectrum of a quasiperiodic Schrödinger operator
cites work: Spectral properties of one dimensional quasi-crystals
cites work: Q5751637
cites work: Spectral properties of a tight binding Hamiltonian with period doubling potential
cites work: Q4284468
cites work: GAP LABELLING THEOREMS FOR ONE DIMENSIONAL DISCRETE SCHRÖDINGER OPERATORS
cites work: JACOBI MATRICES WITH RANDOM POTENTIALS TAKING FINITELY MANY VALUES
cites work: Q3713496
cites work: Q5653193
cites work: Une méthode pour minorer les exposants de Lyapounov et quelques exemples montrant le caractère local d'un théorème d'Arnold et de Moser sur le tore de dimension 2
cites work: Almost periodic Schrödinger operators. II: The integrated density of states
description: scientific article; zbMATH DE number 490659