## Fractal dimensions and the phenomenon of intermittency in quantum dynamics.(English)Zbl 1012.81018

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, I. Guarneri and H. Schulz-Baldes, Math. Phys. Electron. J. 5, Paper No. 1, 16 p. (1999; Zbl 0910.47059)].

### MSC:

 81Q99 General mathematical topics and methods in quantum theory 81Q05 Closed and approximate solutions to the Schrödinger, Dirac, Klein-Gordon and other equations of quantum mechanics 35J10 Schrödinger operator, Schrödinger equation 35Q40 PDEs in connection with quantum mechanics 28A80 Fractals

### Keywords:

fractal dimension; moments of order $$p$$; quantum dynamics

Zbl 0910.47059
Full Text:

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