Mixed lower bounds for quantum transport.
type: Item
instance of: scholarly article
title: Mixed lower bounds for quantum transport.
zbMATH Open document ID: 1060.47070
DOI: 10.1016/S0022-1236(02)00066-6
author: Serguei Tcheremchantsev
published in: Journal of Functional Analysis
publication date: 2003-04-09T00:00:00Z
review text: Given a separable Hilbert space \({\mathcal H}\) and a basis \((e_n)_{n\in{\mathbb N}}\), the abstract position operator \(X_e\) with respect to \((e)\) is defined by \(X_e\psi:=\sum_nn\langle e_n,\psi\rangle e_n\) for all \(\psi\in{\mathcal H}\) with \(\sum_n| n\langle e_n,\psi\rangle| ^2<\infty\). For every \(p\in {\mathbb N}\), the \(p\)th momentum of \(X_e\) is the quadratic form \(\langle X^p_e\rangle\) defined by \(\langle X_e^p\rangle_\psi:=\langle\psi,X_e^p\psi\rangle=\sum_nn^p| \langle e_n,\psi\rangle| ^2\) for all \(\psi\in{\mathcal H}\) with \(\sum_nn^p| \langle e_n,\psi\rangle| ^2<\infty\). Let \(H\) be a self-adjoint operator in \({\mathcal H}\) and choose \(\psi\in{\mathcal H}\) such that the Cesàro mean \(c^p_{e,\psi}(t) :=\frac{1}{t}\int_0^t\,ds\,\langle X^p_e\rangle_{e^{is H}\psi}\) is well-defined for every \(t>0\). The temporal evolution of the growth exponents \(\alpha_{\psi,e}^\pm(p)\) defined by \[ \alpha_{\psi,e}^+(p) :=\limsup_{t\to\infty}\frac{\log(c_{e,\psi}^p(t))}{\log(t)}\quad \text{and}\quad \alpha_{\psi,e}^-(p) :=\liminf_{t\to\infty}\frac{\log(c_{e,\psi}^p(t))}{\log(t)} \] has been the object of recent research. It is known that \(\alpha_{e,\psi}^+(p)/p\) can be estimated from below by the Hausdorff dimension and \(\alpha_{e,\psi}^-(p)/p\) by the packing dimension of the spectral measure \(\mu_\psi\) of \(H\) (see, e.g., \textit{K. Falconer} [Techniques in Fractal Geometry: Mathematical Foundations and Applications (Wiley: New York) (1997; Zbl 0689.28003)] and Appendix A of the present article for definitions). Improvements of these results in recent years have used multifractal dimensions as lower bounds. In contrast to \(\alpha_{e,\psi}^\pm(p)/p\), all these lower bounds do not depend on the basis \((e)\). The present article introduces a further strengthened lower bound that does not merely depend on the spectral measure \(\mu_\psi\), but also on the generalized eigenfunctions of \(H\) and the choice of \((e)\). As it stands, this re…
reviewed by: Bernd Kuckert
Mathematics Subject Classification ID: 47N50
Mathematics Subject Classification ID: 81Q10
Mathematics Subject Classification ID: 47N55
Mathematics Subject Classification ID: 82C70
Mathematics Subject Classification ID: 47B25
zbMATH DE Number: 1895823
zbMATH Keywords: quantum transport
zbMATH Keywords: abstract position operator
zbMATH Keywords: growth exponent
zbMATH Keywords: multifractal dimension
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 1895823