Semiclassical asymptotics of perturbed cat maps

Abstract

We derive an exact representation for tr U n , where U is the quantum propagator associated with an Anosov-perturbed cat map. This takes the form of a sum over the fixed points of the n th iterate of the classical transformation, the contribution of each one being given by an n -fold multiple integral. We focus in particular on the case when n = 1. An asymptotic evaluation of the integral in question then leads to a complete semiclassical series expansion, the first term of which corresponds to the Gutzwiller–Tabor trace formula. It is demonstrated that this series diverges, but that summing it down to its least term provides an approximation to the quantum trace that is exponentially accurate in 1/ ħ . 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 existence of unusual relations between different orbit actions. It also allows us to recover the semiclassical contributions from the complex orbits explicitly, using Borel resummation. These exponentially subdominant terms are shown to exhibit the Stokes phenomenon, which causes them to depend sensitively on the size of the perturbation parameter. Finally, we develop an alternative expansion based on the orbits of the unperturbed cat map. Rather than diverging, this is shown to converge absolutely, thus making possible an exact calculation of the quantum trace using only classical mechanics. Its properties are, however, distinctly anti-semiclassical.