An exact renormalization formula for the Maryland model

Abstract

We discuss the difference Schr\"odinger equation $\psi_{k+1}+\psi_{k-1}+\lambda \cot(\pi\omega k+\theta)\psi_k=E\psi_k$, $k\in\mathbb{Z}$, where $\lambda$, $\omega$, $\theta$ and $E$ are parameters. We obtain explicit renormalization formulas relating its solutions for large $|k|$ to solutions of the equation with new parameters $\lambda$, $\omega$, $\theta$ and $E$ for bounded $|k|$. These formulas are similar to the renormalization formulas from the theory of Gaussian exponential sums.