98-410 Barbaroux J.M., Tcheremchantsev S.

Universal Lower Bounds for Quantum Diffusion (78K, LATeX) Jun 2, 98

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Abstract. We study the connections between dynamical properties of Schr\"o\-din\-ger operators on separable Hilbert space $\gH$ and the properties of corresponding spectral measures. Our main result establish a direct relation between the Fourier transform of spectral measure and the moments of order $p$. This allows us to extend earlier results of Last and Barbaroux, Combes, Montcho based on the Strichartz Theorem. In particular, we obtain $\langle \langle \vert X \vert ^p \rangle _{\psi (t) } \rangle (T) \ge h(T)$, where $h(T)$ is some function that has not necessarily the form $T^{\alpha}$, and we derive lower bounds for $\langle \langle \vert X \vert ^p \rangle _{\psi (t) } \rangle (T_k)$, for subsequences of time $T_k \nearrow +\infty$. We prove also a necessary condition for the dynamical localization in the presence of pure point spectrum. The results are applied to different concrete models of Schr\"odinger operators.

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