Reducibility and point spectrum for linear quasi-periodic skew-products
type: Item
instance of: scholarly article
title: Reducibility and point spectrum for linear quasi-periodic skew-products
zbMATH Open document ID: 0901.34043
published in: Documenta Mathematica
publication date: 1998-08-06T00:00:00Z
full work available at URL: https://eudml.org/doc/225807
full work available at URL: Eliasson.MAN.html
review text: Summary: The author considers linear quasi-periodic skew-product systems on \(\mathbb{T}^d\times G\) where \(G\) is some matrix group. When the quasi-periodic frequencies are Diophantine such systems can be studied by perturbation theory of KAM-type and it is known since the mid 60's that most systems sufficiently close to constant coefficients are reducible, i.e. their dynamics is basically the same as for systems with constant coefficients. In the late 80's a perturbation theory was developed for the other extreme. Fröhlich-Spencer-Wittver and Sinai, independently, were able to prove that certain discrete Schrödinger equations sufficiently far from constant coefficients have pure point spectra, which imply a dynamics completely different from systems with constant coefficients. In recent years these methods are improved and in particular \(\text{SL}(2,\mathbb{R})\) -- related to the Schrödinger equation -- and \(\text{SO}(3,\mathbb{R})\) are well studied.
Mathematics Subject Classification ID: 34C20
Mathematics Subject Classification ID: 81Q05
Mathematics Subject Classification ID: 37A30
Mathematics Subject Classification ID: 34L05
Mathematics Subject Classification ID: 34C27
zbMATH DE Number: 1184287
zbMATH Keywords: quasi-periodicity
zbMATH Keywords: linear quasi-periodic skew-product systems
zbMATH Keywords: perturbation theory
zbMATH Keywords: KAM-type
zbMATH Keywords: discrete Schrödinger equations
zbMATH Keywords: point spectrum
author: L. H. Eliasson
MaRDI profile type: Publication
description: scientific article; zbMATH DE number 1184287