Maslov indices in the Gutzwiller trace formula

Abstract

The Gutzwiller trace formula is a semi-classical approximation for the density of states of a bound quantum system, expressed in terms of a sum over periodic orbits of the corresponding classical system. The authors establish the existence of a topological invariant associated with classical periodic orbits, namely the winding number of their invariant manifolds, and show that this winding number is precisely the index which appears in the trace formula. The proof is valid in any number of dimensions. Explicit formulae for the index are given.