Metamorphosis of a Cantor spectrum due to classical chaos.

Abstract

We study how a Cantor spectrum, its level statistics, and corresponding dynamics are affected by the onset of classical chaos. While the spectrum undergoes visible changes, its level spacing distribution follows an inverse power law p(s)\ensuremath{\sim}${\mathit{s}}^{\mathrm{\ensuremath{-}}3/2}$ on small scales. We find a crossover which is manifested in the time domain by two diffusive regimes characterized by a classical and a quantum-mechanical diffusion coefficient. In the strong quantum limit we show by means of a transformation that the spectrum is governed by the IintegrableP Harper equation, even if the classical phase space is strongly chaotic.