On the nature of the spectrum of the quasi-periodic Schrödinger operator.
type: Item
instance of: scholarly article
title: On the nature of the spectrum of the quasi-periodic Schrödinger operator.
zbMATH Open document ID: 1036.34097
DOI: 10.1016/S1468-1218(01)00012-8
author: Roberta Fabbri
author: Russell Johnson
author: Raffaella Pavani
published in: Nonlinear Analysis. Real World Applications
publication date: 2003-05-25T00:00:00Z
review text: \textit{E. I. Dinaburg} and \textit{Ya. G. Sinai} [Funct. Anal. Appl. 9, 279--289 (1975; Zbl 0333.34014)] were the first to prove the existence of a large class of quasi-periodic potentials \(q\) for which the spectrum of \(H(q):= -d^2/dt^2+ q\) is, in the high-energy regime, absolutely continuous and a Cantor set. \textit{J. Moser} [Comment. Math. Helv. 56, 198--224 (1981; Zbl 0477.34018)] constructed a limit-periodic \(q\) for which the spectrum of \(H(q)\) is a Cantor set and \textit{J. Avron} and \textit{B. Simon} [Commun. Math. Phys. 82, 101--120 (1981; Zbl 0484.35069)] and \textit{V. A. Chulaevskij} [Usp. Mat. Nauk 36, 143--144 (1981; Zbl 0487.34021)] established that this was the generic situation for limit-periodic potentials. Using results of Moser's paper, the authors show that, given \(r\in\mathbb{N}_0\), generically quasi-periodic potentials \(q\in C^r\) give rise to a Lyapunov exponent associated with the eigensolutions of \(H(q)\) which is positive on a set of locally positive measure. The authors' second theorem states that generically the spectrum of \(H(q)\) is a Cantor set for quasi-periodic potentials \(q\in C^0\). This result can already be found in [\textit{R. Johnson}, J. Differ. Equations 91, 88--110 (1991; Zbl 0734.34074)], but was ``not completely proved'' there, as the authors put it. Since further explanation is lacking and the proof given here draws heavily on this paper and is ``sketchy at some points, especially when the necessary arguments are to be found in other papers'', the presentation is less connected and transparent than one might ideally wish. Finally, the authors consider the example \[ q_0(t):= \cos t+ \cos\sqrt{2} t+ \cos\sqrt{3} t \] and provide numerical evidence for the conjecture that the spectrum of \(H(q_0)\) has at least four gaps in the low-energy regime.
Mathematics Subject Classification ID: 34L40
Mathematics Subject Classification ID: 81Q10
Mathematics Subject Classification ID: 34L05
zbMATH DE Number: 1915788
zbMATH Keywords: quasi-periodic Schrödinger operator
zbMATH Keywords: Cantor spectrum
zbMATH Keywords: rotation number
zbMATH Keywords: Lyapunov exponent
reviewed by: Hubert Kalf
describes a project that uses: mctoolbox
MaRDI profile type: Publication
cites work: Q5571891
cites work: Anderson localization for the 1-D discrete Schrödinger operator with two-frequency potential
cites work: Dichotomies in stability theory
cites work: The algebraic-geometric AKNS potentials
cites work: NON-LINEAR EQUATIONS OF KORTEWEG-DE VRIES TYPE, FINITE-ZONE LINEAR OPERATORS, AND ABELIAN VARIETIES
cites work: Floquet solutions for the 1-dimensional quasi-periodic Schrödinger equation
cites work: Discrete one-dimensional quasi-periodic Schrödinger operators with pure point spectrum
cites work: On the Lyapounov exponent of certain SL\((2,\mathbb{R})\)-valued cocycles
cites work: Genericity of exponential dichotomy for two-dimensional differential systems
cites work: Localization for a class of one dimensional quasi-periodic Schrödinger operators
cites work: Q4864704
cites work: Exponential dichotomy, rotation number, and linear differential operators with bounded coefficients
cites work: Cantor spectrum for the quasi-periodic Schrödinger equation
cites work: The rotation number for almost periodic potentials
cites work: Ergodic Properties of Linear Dynamical Systems
cites work: Q3341616
cites work: Q5540675
cites work: An example of a Schrödinger equation with almost periodic potential and nowhere dense spectrum
cites work: Q4379364
cites work: The numerical approximation of the rotation number of planar maps
cites work: Conjugating the Poincaré-map to a rotation
cites work: A spectral theory for linear differential systems
cites work: Loop groups and equations of KdV type
cites work: Anderson localization for one-dimensional difference Schrödinger operator with quasiperiodic potential
full work available at URL: s1468-1218(01)00012-8
OpenAlex ID: W2060179434
description: scientific article; zbMATH DE number 1915788