Universal Lower Bounds for Quantum Diffusion

摘要

We study the connections between dynamical properties of Schrodinger operators H on separable Hilbert space H and the properties of corresponding spectral measures. Our main result establishes a relation for the moment of order p of the form |x|~p>_(#psi#(t))> (T) ident to T~(-1) integral from x=0 to x=T of || |X|~(p/2)e~(-itH)#psi#||~2_H) dt >= L~(#psi#, p/d)(T). Here L~(#psi#, p/d)(T) is a function connected to the behavior of the Fourier transform of measures in the subclass of measures absolutely continuous with respect to the spectral measure #mu#_(#psi#). Beyond the intrinsic interest of the general formulation (1), this result allows us to derive necessary conditions for dynamical localization in the presence of a pure point spectrum. On the other hand, if we focus on subsequences of time T_k arrow up + infinity, we can exhibit lower bounds which are, in certain cases, strictly larger than the well-known power-law lower bound for all T expressed in terms of the Hausdorff dimension of spectral measures.