Extension of the Conley--Zehnder Index and Calculation of the Maslov-Type Index Intervening in Gutzwiller's Trace Formula

TL;DR

This paper extends the Conley--Zehnder index to compute the Gutzwiller--Maslov index for Hamiltonian periodic orbits, providing explicit formulas and linking it to the Maslov index via symplectic topology tools.

Contribution

It introduces a non-trivial extension of the Conley--Zehnder index for calculating the Gutzwiller--Maslov index in Gutzwiller's trace formula, applicable to symplectic paths with arbitrary endpoints.

Findings

Derived explicit formula for the Gutzwiller--Maslov index.

Connected the extended index to the classical Maslov index.

Applicable to symplectic paths with arbitrary endpoints.

Abstract

The aim of this paper is to give an explicit formula for the calculation of the Gutzwiller--Maslov index of a Hamiltonian periodic orbit. We identify the index appearing in Gutzwiller's trace formula with a non-trivial extension of the Conley--Zehnder index. This index can be related to the usual Maslov index using the theory of the metaplectic group and the formalism of the Arnol'd--Leray--Maslov index developed in previous work, and is extended to symplectic paths with arbitrary endpoint.