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Intermittent lower bound on quantum diffusion. (English) Zbl 1001.81019
Let $$H$$ be a self-adjoint, bounded Hamiltonian acting on the Hilbert space $$\ell^2(\mathbb{N})$$. The canonical base is denoted by $$(|n \rangle)_{n \in\mathbb{N}}$$. The unbounded position operator on $$\ell^2(\mathbb{N})$$ is defined by $$X|n\rangle=n|n\rangle$$. The authors are interested in studying the spreading of a wave packet initially localized at $$|0 \rangle$$ under the quantum dynamics generated by $$H$$, and analyze it with help of the moments of the time-averaged probability distribution on $$\mathbb{N}$$, notably the time-averaged expectation values of powers of $$X$$. The authors’ goal is to characterize the spreading by properties of the spectral measure $$\mu$$ of $$H$$ with respect to $$|0\rangle$$.

##### MSC:
 81Q10 Selfadjoint operator theory in quantum theory, including spectral analysis 37N20 Dynamical systems in other branches of physics (quantum mechanics, general relativity, laser physics) 28A80 Fractals
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