Semiclassical singularities from bifurcating orbits

Abstract

We study how the singular behaviour of classical systems at bifurcations is reflected by their quantum counterpart. The semiclassical contributions of individual periodic orbits to trace formulae of Gutzwiller type are known to diverge when orbits bifurcate, a situation characteristic for systems with a mixed phase space. This singular behaviour is reflected by the quantum system in the semiclassical limit, in which the individual contributions remain valid closer to a bifurcation, while the true collective amplitude at the bifurcation increases with some inverse power of Planck's constant (with an exponent depending on the type of bifurcation). We illustrate the interplay between the two competing limits (closeness to a bifurcation and smallness of $\hbar$) numerically for a generic dynamical system, the kicked top.