Renormalized oscillation theory for linear Hamiltonian systems on [0,1] via the Maslov index

TL;DR

This paper develops a renormalized oscillation theory for linear Hamiltonian systems on [0,1], utilizing the Maslov index to unify and extend classical oscillation results to systems including Dirac, Sturm-Liouville, and differential-algebraic equations.

Contribution

It introduces a natural approach to oscillation theory using the Maslov index, applicable to a broad class of Hamiltonian systems with nonlinear spectral parameters.

Findings

Established a Maslov index framework for oscillation analysis

Included Dirac and Sturm-Liouville systems in the applicability class

Extended results to differential-algebraic systems with nonlinear spectral parameters

Abstract

Working with a general class of linear Hamiltonian systems on $[0, 1]$, we show that renormalized oscillation results can be obtained in a natural way through consideration of the Maslov index associated with appropriately chosen paths of Lagrangian subspaces of $\mathbb{C}^{2n}$. We verify that our applicability class includes Dirac and Sturm-Liouville systems, as well as a system arising from differential-algebraic equations for which the spectral parameter appears nonlinearly.