Commutation relations for unitary operators. I
type: Item
instance of: scholarly article
title: Commutation relations for unitary operators. I
zbMATH Open document ID: 1330.81090
published in: Journal of Functional Analysis
publication date: 2015-08-26T00:00:00Z
review text: There is considered a class of unitary operators \(U\), called propagating unitary operators, satisfying a certain commutation relation with an auxiliary selfadjoint operator \(A\). One main result (Theorem 2.3) is: If \(\Theta\) is a subset of the torus \(\mathbb{R}/2\pi\mathbb{Z}\), then \(e^{i\Theta}\cup \sigma(U)\) consists only of a finite number of eigenvalues with finite multiplicity. Certain matrix elements of \((\mathbf{1}-z^{-1}U^*)^{-1}\) and \((\mathbf{1}-\overline{z}^{-1}U^*)^{-1}\) are continuously extendable to \(e^{i\Theta}\setminus \sigma_{pp}(U)\). Moreover it is proved that \(e^{i\Theta}\cap \sigma_{sc}(U)=\emptyset\). The proof of this theorem given in Section 4 uses the Virial theorem and Mourre inequalities. These abstract results are applied to Floquet operators associated to AC-Stark perturbations of the harmonic oscillator on \(L^2(\mathbb{R})\). New results are contained in Theorems 3.1 and 3.2. They show the absence of singular continuous spectra for Floquet operators associated to time-dependent Hamiltonians perturbed by certain multiplication operators. The remaining sections treat regularity classes, Weyl commutation relations and \(U\)-smoothness in a little more detail. Part II has appeared in [J. Approx. Theory 199, Article ID 5016, 63--94 (2015; Zbl 1333.81145)].
reviewed by: Michael Demuth
Mathematics Subject Classification ID: 81Q10
Mathematics Subject Classification ID: 81Q05
Mathematics Subject Classification ID: 81S05
Mathematics Subject Classification ID: 47B25
zbMATH DE Number: 6475680
zbMATH Keywords: spectrum
zbMATH Keywords: commutator
zbMATH Keywords: unitary operator
author: M. A. Astaburuaga
author: O. Bourget
author: V. H. Cortés
MaRDI profile type: Publication
full work available at URL: j.jfa.2015.01.011
OpenAlex ID: W4206534666
DOI: 10.1016/J.JFA.2015.01.011
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description: scientific article; zbMATH DE number 6475680