Spectral of quasienergies
type: Item
instance of: scholarly article
title: Spectral of quasienergies
zbMATH Open document ID: 0897.46015
author: Lyn III Geisler
author: James S. Howland
published in: Journal of Mathematical Analysis and Applications
publication date: 1998-07-28T00:00:00Z
review text: Let \(B\) be the closed unit ball in the space \(C(T^1)^\ast\) of measures on the unit circle \(T^1\). So \(B\) is a complete metric space in the \(w^\ast\)-topology. Let \(\Sigma_{\text{ac}}\) (resp. pp, sc, ps, and pc) be the set of measures in \(B\) which are absolutely continuous (resp. pure point, singular continuous, purely singular, and purely continuous). The fact is used that \(\Sigma_{\text{pc}}\) and \(\Sigma_{\text{ps}}\) are both \(G_\delta\) subsets of \(B\). One shows that if \(\mathcal H\) is a complete metric space and \(F:\mathcal H\to B\) is continuous such that both \(F^{-1}(\Sigma_{\text{pp}})\) and \(F^{-1}(\Sigma_{\text{ac}})\) are dense then \(F^{-1}(\Sigma_{\text{sc}})\) is a dense \(G_\delta\) subset of \(\mathcal H\). This result is applied to the kicked rotor with the time-periodic Hamiltonian \(H(t)=-\alpha d^2/d\theta^2 + v(\theta) \sum_{n=-\infty}^\infty\delta(t-2\pi n)\) in \(L_2(d\theta/2\pi,T^1)\), and to the quasiperiodic Rabi oscillator represented by the operator \(K(\alpha,H)=i\alpha_1\partial/\partial\theta_1+ i\alpha_2\partial/\partial\theta_2 + H(\theta)\) where \(H(\theta)\) is a smooth \(2\times 2\) Hermitian matrix-valued function on the torus \(T^2\). In the first case one proves that (1) the Floquet operator \(U(\alpha,v)\) corresponding to \(H(t)\) is purely continuous for a generic set of parameters \(\alpha\) and \(v\) analytic (generic means dense and \(G_\delta\) in an appropriate complete metric space), and (2) \(U(\alpha,v)\) is purely singular continuous for a generic set of \((\alpha,v)\), with \(v\) of bounded variation. In the second case one proves that \(K(\alpha,H)\) is purely continuous for a generic set of \((\alpha,H)\).
Mathematics Subject Classification ID: 46E27
Mathematics Subject Classification ID: 47A56
Mathematics Subject Classification ID: 47N50
Mathematics Subject Classification ID: 81Q05
zbMATH DE Number: 1005687
zbMATH Keywords: Floquet operator
zbMATH Keywords: kicked rotor
zbMATH Keywords: complete metric space
zbMATH Keywords: \(w^*\)-topology
zbMATH Keywords: time-periodic Hamiltonian
zbMATH Keywords: quasiperiodic Rabi oscillator
zbMATH Keywords: smooth \(2\times 2\) Hermitian matrix-valued function
reviewed by: P. Št'ovíček
MaRDI profile type: Publication
full work available at URL: jmaa.1997.5277
OpenAlex ID: W2036456659
cites work: Q3704322
cites work: Floquet spectrum for two-level systems in quasiperiodic time-dependent fields
cites work: Non-recurrent behaviour in quantum dynamics
cites work: Q3134909
cites work: Spectral and stability aspects of quantum chaos
cites work: Operators with singular continuous spectrum. I: General operators
DOI: 10.1006/JMAA.1997.5277
description: scientific article; zbMATH DE number 1005687