An exact renormalization formula for the Maryland model

Abstract

We discuss the difference Schrödinger 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.

15 pages

References in corpus (1)

• Pointwise existence of the Lyapunov exponent for a quasi-periodic equation

Cited by in corpus (3)

• Arithmetic Spectral Transitions for the Maryland Model
• Pure point spectrum for the Maryland model: a constructive proof
• A series of spectral gaps for the almost Mathieu operator with a small coupling constant