Universal Lower Bounds for Quantum Diffusion

We study the connections between dynamical properties of Schrödinger 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〉ψ(t)〉(T)≡T−1 ∫T0 ‖|X|p/2 e−itHψ‖2H dt⩾Lψ, p/d(T). (1) Here Lψ, 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 μψ. 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 Tk↗+∞, 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.