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Universal Lower Bounds for Quantum Diffusion

✍ Scribed by J.M. Barbaroux; S. Tcheremchantsev

Publisher: Elsevier Science
Year: 1999
Tongue: English
Weight: 220 KB
Volume: 168
Category: Article
ISSN: 0022-1236
DOI: 10.1006/jfan.1999.3471

No coin nor oath required. For personal study only.

✦ Synopsis

We study the connections between dynamical properties of Schro dinger 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

H dt L , pÂd (T ).

(1)

Here L , pÂd (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 + . 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 Z+ , 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.

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