Barbaroux J.M., Tcheremchantsev S.
Universal Lower Bounds for Quantum Diffusion
(78K, LATeX)

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.