Relative Oscillation Theory for Jacobi Matrices Extended

TL;DR

This paper extends relative oscillation theory for finite Jacobi matrices to include arbitrary perturbations, providing a comprehensive framework that relates eigenvalue differences to weighted sign-changes of solutions' Wronskians.

Contribution

It generalizes existing results from diagonal perturbations to all perturbations, simplifies proofs, and establishes a new comparison theorem for spectral intervals.

Findings

Eigenvalue differences correspond to weighted sign-changes of Wronskians.

Extended theory to arbitrary perturbations beyond diagonal cases.

Simplified proof and established comparison theorem.

Abstract

We present a comprehensive treatment of relative oscillation theory for finite Jacobi matrices. We show that the difference of the number of eigenvalues of two Jacobi matrices in an interval equals the number of weighted sign-changes of the Wronskian of suitable solutions of the two underlying difference equations. Until now only the case of perturbations of the main diagonal was known. We extend the known results to arbitrary perturbations, allow any (half-)open and closed spectral intervals, simplify the proof, and establish the comparison theorem.