Abstract

(joint work with A.H. Werner, C. Cedzich, D. Meschede, A. Alberti, T. Rybar. See arXiv:1302.2094 and arXiv:1302.2081, resp. PRL 111 , 160601 and 110, 190601)

I present a very simple quantum system, whose long time behavior depends extremely sensitively on a parameter E. (1) For rational E one sees some revivals (exponentially sharp in the denominator of E) followed ultimately by ballistic expansion. (2) For typical E (Lebesgue- almost all) one has localization with exponentially localized eigenfunctions, but there is also (3) a dense set of E for which one has hierarchical motion: An infinite hierarchy of time scales on each of which one has sharper and sharper revivals (with a repetition of everything before that) followed by larger and larger recursions. The spectrum of the walk operator is absolutely continuous, pure point, and singular continuous in these three cases. We also explain how on a fixed finite time scale these distinctions become irrelevant and it is enough to know an appropriate initial segment of the continued fraction expansion of E.