On the Measure of the Absolutely Continuous Spectrum for Jacobi Matrices

TL;DR

This paper uses classical approximation theory to analyze the measure of the absolutely continuous spectrum of Jacobi matrices, providing bounds and generalizations of existing inequalities.

Contribution

It introduces new bounds on the measure of the absolutely continuous spectrum and generalizes the differential inequality for the integrated density of states.

Findings

Upper bound on the measure of the absolutely continuous spectrum.

Bound incorporates the value distribution of diagonal elements.

Generalized differential inequality for the integrated density of states.

Abstract

We apply the methods of classical approximation theory (extreme properties of polynomials) to study the essential support $\Sigma_{ac}$ of the absolutely continuous spectrum of Jacobi matrices. First, we prove an upper bound on the measure of $\Sigma_{ac}$ which takes into account the value distribution of the diagonal elements, and implies the bound due to Deift-Simon and Poltoratski-Remling. Second, we generalise the differential inequality of Deift-Simon for the integrated density of states associated with the absolutely continuous spectrum to general Jacobi matrices.