SEMICLASSICAL ASYMPTOTICS OF PERTURBED CAT MAPS

We derive an exact representation for trU(n), where U is the quantum p ropagator associated with an Anosov-perturbed cat map. This takes the form of a sum over the fixed points of the nth iterate of the classica l transformation, the contribution of each one being given by an n-fol d multiple integral. We focus in particular on the case when n = 1. An asymptotic evaluation of the integral in question then leads to a com plete semiclassical series expansion, the first term of which correspo nds to the Gutzwiller-Tabor trace formula. It is demonstrated that thi s series diverges, but that summing it down to its least term provides an approximation to the quantum trace that is exponentially accurate in 1/HBAR. A simple, universal approximation to the late terms is then derived. This explains the divergence of the semiclassical expansion in terms of complex (tunnelling) periodic orbits, and implies the exis tence of unusual relations between different orbit actions. It also al lows us to recover the semiclassical contributions from the complex or bits explicitly, using Borel resummation. These exponentially subdomin ant terms are shown to exhibit the Stokes phenomenon, which causes the m to depend sensitively on the size of the perturbation parameter. Fin ally, we develop an alternative expansion based on the orbits of the u nperturbed cat map. Rather than diverging, this is shown to converge a bsolutely, thus making possible an exact calculation of the quantum tr ace using only classical mechanics. Its properties are, however, disti nctly anti-semiclassical.