Morse index and Maslov-type index of the discrete Hamiltonian system

TL;DR

This paper introduces a Maslov-type index for discrete Hamiltonian systems, relates it to Morse index, and extends index iteration theories to this discrete setting, providing a broader framework for analyzing such systems.

Contribution

It defines a new Maslov-type index for discrete Hamiltonian systems and generalizes existing Morse index relations to cases where omegadiffer from 1, extending index theories.

Findings

Established the relation between Morse and Maslov-type indices for discrete systems.

Proved well-posedness of splitting numbers for the discrete Hamiltonian system.

Extended index iteration theories to the discrete Hamiltonian system case.

Abstract

In this paper, we give the definition of Maslov-type index of the discrete Hamiltonian system, and obtain the relation of Morse index and Maslov-type index of the discrete Hamiltonian system which is a generalization of case $\omega=1$ in \cite{RoS1}, \cite{RoS2} and \cite{Maz1} to case $\omega \in {\bf U}$ via direct method which is different from that of \cite{RoS1}, \cite{RoS2} and \cite{Maz1}. Moreover well-posedness of the splitting numbers $S_{\omega}$ of the discrete Hamiltonian system is proven, thus index iteration theories in \cite{Bot1} and \cite{Lon4} are also valid for the discrete Hamiltonian system case.