Using invariance by fixed-endpoints homotopies and a generalized notion of symplectic Cayley transform, we prove a product formula for the Conley–Zehnder index of continuous paths with arbitrary endpoints in the symplectic group. We discuss two applications of the formula, to the metaplectic group and to periodic solutions of Hamiltonian systems.

In this series of eight papers we present the applications of methods from wavelet analysis to polynomial approximations for a number of accelerator physics problems. In this part we consider application of FWT to metaplectic representation(quantum and chaotical problems) and quasiclassics.

Historically θ-series have been one of the major methods of constructing automorphic forms. A representation-theoretic appoach to the theory of θ-series, as discoved by A. Weil [18] and extended by R. Howe [12], is based on the oscillator representation of the metaplectic group (cf. [17] for a recent survey). In this paper we propose a geometric interpretation this representation (in the nonram...