Spectral bounds for ergodic Jacobi operators

TL;DR

This paper establishes bounds on the spectrum of ergodic Jacobi operators using Lyapunov exponents, leveraging potential theory to connect spectral properties with dynamical systems.

Contribution

It introduces new spectral bounds for ergodic Jacobi operators based on Lyapunov exponents and potential theory, advancing understanding of their spectral structure.

Findings

Bounds on the Lebesgue measure of the spectrum

Estimates on the spectral extremal points

Connections between Lyapunov exponents and spectral gaps

Abstract

We consider ergodic Jacobi operators and obtain estimates on the Lebesgue measure and the distance between maximum and minimum points of the spectrum in terms of the Lyapunov exponent. Our proofs are based on results from logarithmic potential theory and their connections with spectral theory of Jacobi operators.