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

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.