Universal lower bounds for quantum diffusion
type: Item
instance of: scholarly article
title: Universal lower bounds for quantum diffusion
zbMATH Open document ID: 0961.81008
author: Jean-Marie Barbaroux
author: Serguei Tcheremchantsev
published in: Journal of Functional Analysis
publication date: 2000-03-27T00:00:00Z
review text: Let \(H\) be a self-adjoint operator in \({\mathcal H}=L^2 (\mathbb{R}^d)\) or in \({\mathcal H}= \ell^2(\mathbb{Z}^d)\). The authors study the asymptotic behaviour of the Cesaro mean \(\langle\langle|X|^p \rangle_{\psi(t)} \rangle(T)\) at time \(T\) of the moments of order \(p\) for \(\psi \in{\mathcal H}\). With the aid of the so-called double-space method an upper bound for \[ B_\psi (T,N):= T^{-1} \int^T_0 \bigl\|\psi_N(t) \bigr\|^2_{\mathcal H} dt \] is proved, where \(\psi_N(t) =F_{<N}e^{-itH} \psi\) and \(F_{<N}\) is the characteristic function of the closed ball of radius \(N\). This key estimate holds without assuming any continuity properties of the spectral measure \(\mu_\psi\) and yields lower bounds for \(\langle\langle|X|^p \rangle_{ \psi(t)} \rangle (T)\) in terms of the Fourier transform of \(\mu_\psi\). The obtained estimates are applied to derive necessary conditions for dynamic localization, provided \(H\) has pure point spectrum. In an appendix the converse of Strichartz' theorem on Fourier asymptotics of uniformly \(\alpha\)-Hölder continuous measures is proved.
reviewed by: André Noll
Mathematics Subject Classification ID: 81Q10
Mathematics Subject Classification ID: 47N50
Mathematics Subject Classification ID: 42A38
zbMATH DE Number: 1389440
zbMATH Keywords: Hölder continuous measures
zbMATH Keywords: self-adjoint operator
zbMATH Keywords: asymptotic behaviour
zbMATH Keywords: Cesaro mean
zbMATH Keywords: Fourier transform
zbMATH Keywords: dynamic localization
zbMATH Keywords: converse of Strichartz' theorem
zbMATH Keywords: Fourier asymptotics
MaRDI profile type: Publication
full work available at URL: jfan.1999.3471
OpenAlex ID: W1989982438
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DOI: 10.1006/JFAN.1999.3471
description: scientific article; zbMATH DE number 1389440