Statistics of resonances and of delay times in quasiperiodic Schr"odinger equations

Abstract

We study the statistical distributions of the resonance widths $\mathcal{P}(\Gamma )$, and of delay times $\mathcal{P}(\tau )$ in one dimensional quasi-periodic tight-binding systems with one open channel. Both quantities are found to decay algebraically as ${\Gamma }^{-\alpha }$, and ${\tau }^{-\gamma }$ on small and large scales respectively. The exponents $\alpha$, and $\gamma$ are related to the fractal dimension $D_0^E$ of the spectrum of the closed system as $\alpha =1+D_0^E$ and $\gamma =2-D_0^E$. Our results are verified for the Harper model at the metal-insulator transition and for Fibonacci lattices.

4 pages, 3 figures, submitted to Phys. Rev. Lett