Sturm intersection theory for periodic Jacobi matrices and linear Hamiltonian systems

TL;DR

This paper extends Sturm-Liouville oscillation theory to periodic Jacobi matrices with matrix entries and linear Hamiltonian systems, using intersection theory to simplify eigenvalue problem analysis.

Contribution

It introduces a unified approach applying intersection theory to both periodic Jacobi matrices and linear Hamiltonian systems, simplifying existing proofs.

Findings

Simplified proof of Sturm-Liouville oscillation theory for matrix-valued Jacobi operators.

Application of intersection theory to eigenvalue problems in Hamiltonian systems.

Clarification of the use of Bott, Maslov, and Conley-Zehnder intersection theories.

Abstract

Sturm-Liouville oscillation theory for periodic Jacobi operators with matrix entries is discussed and illustrated. The proof simplifies and clarifies the use of intersection theory of Bott, Maslov and Conley-Zehnder. It is shown that the eigenvalue problem for linear Hamiltonian systems can be dealt with by the same approach.