Slow decay of temporal correlations in quantum systems with Cantor spectra

We prove that the temporal autocorrelation function C(t) for quantum systems with Cantor spectra has an algebraic decay C(t){~}t^-δ, where {δ} equals the generalized dimension D₂ of the spectral measure and is bounded by the Hausdorff dimension D₀{≥}{δ}. We study various incommensurate systems with singular continuous and absolutely continuous Cantor spectra and find extremely slow correlation decays in singular continuous cases ({δ}=0.14 for the critical Harper model and 0{≤}0.84 for the Fibonacci chains). In the kicked Harper model we deomonstrate that the quantum mechanical decay is unrelated to the existence of classical chaos.