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

Abstract

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