Eigenfunction Expansion and the Decomposition of Jacobi Operators on $\mathbb{Z}$

TL;DR

This paper develops an eigenfunction expansion for Jacobi operators on the integer lattice, enabling their decomposition into simpler components, which advances spectral analysis techniques for such operators.

Contribution

It introduces a new eigenfunction expansion theorem for the singular part of Jacobi operators on b, facilitating their decomposition into direct integrals of half-line operators.

Findings

Eigenfunction expansion for the singular part of Jacobi operators

Decomposition of Jacobi operators into direct integrals of half-line operators

Application of subordinate solutions to spectral analysis

Abstract

Let $J$ be a Jacobi operator on $\ell^2\left(\mathbb{Z}\right)$. We prove an eigenfunction expansion theorem for the singular part of $J$ using subordinate solutions to the eigenvalue equation. We exploit this theorem in order to show that $J$ can be decomposed as a direct integral of half-line operators.