ON THE DYNAMICAL MEANING OF SPECTRAL DIMENSIONS

Dynamical Localization theory has drawn attention to general spectral conditions which make quantum wave packet diffusion possible, and it w as found that dimensional properties of the Local Density of States pl ay a crucial role in that connection. In this paper an abstract result in this vein is presented. Time averaging over the trajectory of a wa vepacket up to time T defines a statistical operator (density matrix). The corresponding entropy increases with time proportional to log T: and the coefficient of proportionality is the Hausdorff dimension of t he Local Density of States, at least if the latter has good scaling pr operties. In more general cases, we give spectral upper and lower boun ds for the increase of entropy. (C) Elsevier, Paris.