Oscillation numbers for continuous Lagrangian paths and Maslov index

TL;DR

This paper develops a comprehensive theory linking oscillation numbers of continuous Lagrangian paths with Maslov index, utilizing symplectic matrix angles and comparative index theory, with applications to Hamiltonian systems.

Contribution

It introduces a novel connection between oscillation numbers and Lidskii angles, extending the theory of Maslov index for continuous Lagrangian paths.

Findings

Established a formula for Maslov index via Lidskii angles.

Connected oscillation numbers with symplectic matrix properties.

Provided comparison theorems for oscillation numbers of Lagrangian paths.

Abstract

In this paper we present the theory of oscillation numbers and dual oscillation numbers for continuous Lagrangian paths in $\mathbb{R}^{2n}$. Our main results include a connection of the oscillation numbers of the given Lagrangian path with the Lidskii angles of a special symplectic orthogonal matrix. We also present Sturmian type comparison and separation theorems for the difference of the oscillation numbers of two continuous Lagrangian paths. These results, as well as the definition of the oscillation number itself, are based on the comparative index theory (Elyseeva, 2009). The applications of these results are directed to the theory of Maslov index of two continuous Lagrangian paths. We derive a formula for the Maslov index via the Lidskii angles of a special symplectic orthogonal matrix, and hence we express the Maslov index as the oscillation number of a certain transformed Lagrangian path. The results and methods are based on a generalization of the recently introduced oscillation numbers and dual oscillation numbers for conjoined bases of linear Hamiltonian systems (Elyseeva, 2019 and 2020) and on the connection between the comparative index and Lidskii angles of symplectic matrices (\v{S}epitka and \v{S}imon Hilscher, 2020).