Singular components of spectral measures for ergodic Jacobi matrices

TL;DR

This paper proves that for ergodic 1D Jacobi operators, the singular parts of spectral measures are almost surely disjoint on the set of positive Lyapunov exponent, providing new insights into spectral theory and the extended Harper's equation.

Contribution

It establishes the mutual disjointness of singular spectral components for ergodic Jacobi matrices and proves Thouless' formula in the dual regions of extended Harper's equation.

Findings

Singular spectral measures are mutually disjoint on positive Lyapunov exponent sets.

First rigorous proof of Thouless' formula for extended Harper's equation in dual regions.

Advances understanding of spectral measures in ergodic Jacobi operators.

Abstract

For ergodic 1d Jacobi operators we prove that the random singular components of any spectral measure are almost surely mutually disjoint as long as one restricts to the set of positive Lyapunov exponent. In the context of extended Harper's equation this yields the first rigorous proof of the Thouless' formula for the Lyapunov exponent in the dual regions.