Quantum chaology of fractal spectra

A few years ago we found the existence of an unbounded quantum mechanical diffusion process which strongly contrasted with the dynamical localization of the kicked rotator and other previously studied quantum systems. Here we review how this phenomenon is related to multifractal properties of the spectrum, its level statistics, and to the algebraic decay of correlations. The Harper model, a single-band description of Block electrons in magnetic fields, shows all these fascinating properties. Classically, however, the Harper model is integrable and thus fails for lateral surface superlattices where chaotic trajectories prevail near the classical limit. We therefore derive a new model, which is reminiscent of Harper''s equation, but is exact under the most general conditions and exhibits chaos in the classical limit. We study the influence of classical chaos on the fractal spectrum known as Hofstadter''s butterfly and make predictions on its observability in lateral surface superlattices. (author)