Renormalized Oscillation Theory for Singular Linear Hamiltonian Systems

TL;DR

This paper extends oscillation theory to singular linear Hamiltonian systems by using the Maslov index, providing a natural framework for analyzing systems with singular boundary conditions.

Contribution

It introduces a renormalized oscillation approach for singular systems via the Maslov index, generalizing previous results for regular systems.

Findings

Established a connection between Maslov index and oscillation counts for singular systems

Extended oscillation theory to systems with singular boundary conditions

Provided a natural framework for analyzing singular Hamiltonian systems

Abstract

Working with a general class of linear Hamiltonian systems with at least one singular boundary condition, 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 $\mathbb{C}^{2n}$. This extends previous work by the authors for regular linear Hamiltonian systems.