Kotani Theory for ergodic matrix-like Jacobi operators

TL;DR

This paper extends Kotani Theory to ergodic matrix-like Jacobi operators, linking zero Lyapunov exponents to absolutely continuous spectrum and providing a Thouless Formula for this class.

Contribution

It generalizes Kotani Theory to matrix-like Jacobi operators with ergodic dynamics, establishing spectral measure relations and a new Thouless Formula.

Findings

Essential closure of zero Lyapunov exponents set matches absolutely continuous spectrum of given multiplicity.

Odd spectrum multiplicities are almost surely empty.

A Thouless Formula is derived for this class of operators.

Abstract

We extend the so-called Kotani Theory for a particular class of ergodic matrix-like Jacobi operators defined in $l^{2}(\mathbb{Z}; \mathbb{C}^{l})$ by the law $[H_{\omega} \textbf{u}]_{n} := D^{*}(T^{n - 1}\omega) \textbf{u}_{n - 1} + D(T^{n}\omega) \textbf{u}_{n + 1} + V(T^{n}\omega) \textbf{u}_{n}$, where $T: \Omega \rightarrow \Omega$ is an ergodic automorphism in the measure space $(\Omega, \nu)$, the map $D: \Omega \rightarrow GL(l, \mathbb{R})$ is bounded, and for each $\omega\in\Omega$, $D(\omega)$ is symmetric. Namely, it is shown that for each $r\in\{1,\ldots,l\}$, the essential closure of $\mathcal{Z}_{r} := \{x \in \mathbb{R}\mid$ exactly $2r$ Lyapunov exponents of $A_z$ are zero$\}$ coincides with $\sigma_{ac,2r}(H_{\omega})$, the absolutely continuous spectrum of multiplicity $2r$, where $A_z$ is a Schr\"odinger-like cocycle induced by $H_\omega$. Moreover, if $k\in\{1,\ldots,2l\}$ is odd, then $\sigma_{ac,k}(H_{\omega})=\emptyset$ for $\nu$-a.e. $\omega\in\Omega$. We also provide a Thouless Formula for such class of operators.