Renormalization-group analysis of the discrete quasiperiodic Schrödinger equation

Abstract

Recently developed scaling concepts in the theory of quasiperiodic dynamical systems are used to develop an exact renormalization group applicable to the discrete, quasiperiodic Schr\"odinger equation. To illustrate the power of the method, we calculate the universal scaling properties of the states and eigenvalue spectrum at and below the localization transition for an energy which corresponds to an integrated density of states of $\frac{1}{2}$. The modulating potential has a frequency $\frac{1}{2}(\sqrt{5}\ensuremath{-}1)$ relative to the underlying lattice for the example we work out in greatest detail.