Renormalized oscillation theory for regular linear non-Hamiltonian systems

TL;DR

This paper extends renormalized oscillation theory to regular linear non-Hamiltonian systems using a generalized Maslov index, enabling monotonicity of crossing points and broadening oscillation-based eigenvalue analysis.

Contribution

It introduces a natural extension of renormalized oscillation theory to non-Hamiltonian systems via a generalized Maslov index, ensuring crossing monotonicity.

Findings

Establishes monotonicity of crossing points in non-Hamiltonian systems.

Demonstrates applicability of the generalized theory to eigenvalue bounds.

First extension of renormalized oscillation approach to non-Hamiltonian systems.

Abstract

In recent work, Baird et al. have generalized the definition of the Maslov index to paths of Grassmannian subspaces that are not necessarily contained in the Lagrangian Grassmannian [T. J. Baird, P. Cornwell, G. Cox, C. Jones, and R. Marangell, {\it Generalized Maslov indices for non-Hamiltonian systems}, SIAM J. Math. Anal. {\bf 54} (2022) 1623-1668]. Such an extension opens up the possibility of applications to non-Hamiltonian systems of ODE, and Baird and his collaborators have taken advantage of this observation to establish oscillation-type results for obtaining lower bounds on eigenvalue counts in this generalized setting. In the current analysis, the author shows that renormalized oscillation theory, appropriately defined in this generalized setting, can be applied in a natural way, and that it has the advantage, as in the traditional setting of linear Hamiltonian systems, of ensuring monotonicity of crossing points as the independent variable increases for a wide range of system/boundary-condition combinations. This seems to mark the first effort to extend the renormalized oscillation approach to the non-Hamiltonian setting.