Uniform approximations for non-generic bifurcation scenarios including bifurcations of ghost orbits

Gutzwiller's trace formula allows interpretation of the density of states o f a classically chaotic quantum system in terms of classical periodic orbit s. It diverges when periodic orbits undergo bifurcations and must be replac ed with a uniform approximation in the vicinity of the bifurcations. As a c haracteristic feature, these approximations require the inclusion of comple x "ghost orbits". By studying an example taken from the Diamagnetic Kepler Problem, viz. the period-quadrupling of the balloon orbit, we demonstrate t hat these ghost orbits themselves can undergo bifurcations, giving rise to non-generic complicated bifurcation scenarios. We extend classical normal f orm theory so as to yield analytic descriptions of both bifurcations of rea l orbits and ghost orbit bifurcations. We then show how the normal form ser ves to obtain a uniform approximation taking the ghost orbit bifurcation in to account. We find that the ghost bifurcation produces signatures in the s emiclassical spectrum in much the same way as a bifurcation of real orbits does. (C) 1999 Academic Press.