Fractal dimensions and the phenomenon of intermittency in quantum dynamics (Q1847872)

Let us consider a separable Hilbert space \(\mathcal H\), an orthonormal basis \(\{e_n\}_{n=1}^{\infty}\) and a self-adjoint operator \(H\). Let \(\psi_t=e^{-itH}\psi\) be the solution of the Schrödinger equation with initial state \(\psi\). With \(X=\sum_nn(\cdot,e_n)e_n\) being the position operator the time-averaged moments of order \(p\) for \(\psi\) are defined by \[ \langle\langle|X|^p\rangle\rangle_{\psi,T}=\frac{1}{T}\int^T_0\||X|^{p/2}e^{-itH}\psi\|^2_{\mathcal H} dt. \] Let \[ \alpha^-(\psi,p,d)=\liminf_{T\to\infty}\frac{\log\langle\langle|X|^p\rangle\rangle_{\psi,T}}{\log T}, \qquad \alpha^+(\psi,p,d)=\limsup_{T\to\infty}\frac{\log\langle\langle|X|^p\rangle\rangle_{\psi,T}}{\log T}. \] The main result of the paper is the following: for any self-adjoint operator \(H\) and for any initial state \(\psi\) such that the associated spectral measure \(d\mu_{\psi}\) satisfies \(D^{\pm}_{\mu_{\psi}}(s)<\infty\) for any \(s\in(0,1)\), then \(\alpha^{\pm}(\psi,p,d)\geq D^{\pm}_{\mu_{\psi}}(1/(1+p/d))p/d\), where \(D^{\pm}_{\mu_{\psi}}(q)\) are \(q\)th generalized fractal dimensions. This result improves results obtained in terms of Hausdorff and packing dimensions [see, for example, \textit{I. Guarneri} and \textit{H. Schulz-Baldes}, Math. Phys. Electron. J. 5, Paper No. 1, 16 p. (1999; Zbl 0910.47059)].