On Direct Integral Expansion for Periodic Block-Operator Jacobi Matrices and Applications

TL;DR

This paper develops a functional model using direct integral expansion to analyze the spectra of periodic block-operator Jacobi matrices, providing bounds on spectral measures and examples with multiple spectral gaps.

Contribution

It introduces a novel functional model for periodic block-operator Jacobi matrices and derives optimal bounds on their spectral Lebesgue measure.

Findings

Derived an upper bound for the Lebesgue measure of spectra

Constructed examples with multiple spectral gaps

Analyzed spectra of 2D partial difference operators

Abstract

We construct a functional model (direct integral expansion) and study the spectra of certain periodic block-operator Jacobi matrices, in particular, of general 2D partial difference operators of the second order. We obtain the upper bound, optimal in a sense, for the Lebesgue measure of their spectra. The examples of the operators for which there are several gaps in the spectrum are given.