Relative Oscillation Theory for Jacobi Matrices

TL;DR

This paper introduces a new relative oscillation theory for Jacobi matrices that compares eigenvalue counts between two matrices using weighted nodes of Wronskians, offering a novel approach to spectral analysis.

Contribution

It develops a relative oscillation framework for Jacobi matrices, replacing traditional eigenvalue counting with a method based on weighted nodes of Wronskians of solutions.

Findings

Provides a new method for comparing eigenvalues of two Jacobi matrices.

Replaces node counting with weighted nodes of Wronskians.

Enhances spectral analysis techniques for Jacobi matrices.

Abstract

We develop relative oscillation theory for Jacobi matrices which, rather than counting the number of eigenvalues of one single matrix, counts the difference between the number of eigenvalues of two different matrices. This is done by replacing nodes of solutions associated with one matrix by weighted nodes of Wronskians of solutions of two different matrices.