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)∼${\mathit{t}}^{\mathrm{-}\mathrm{\delta }}$, where δ equals the generalized dimension ${\mathit{D}}_2$ of the spectral measure and is bounded by the Hausdorff dimension ${\mathit{D}}_0$≥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<δ≤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.