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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 \(|0\rangle\).

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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