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Mixed lower bounds for quantum transport. (English) Zbl 1060.47070
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., 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 result is much less appealing than the earlier ones, in particular due to its dependence of \((e)\). But using it as a tool, reformulations and further improvements of the earlier results are eventually obtained.

MSC:
47N50 Applications of operator theory in the physical sciences
81Q10 Selfadjoint operator theory in quantum theory, including spectral analysis
47N55 Applications of operator theory in statistical physics (MSC2000)
82C70 Transport processes in time-dependent statistical mechanics
47B25 Linear symmetric and selfadjoint operators (unbounded)
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