Ghost Orbit Bifurcations in Semiclassical Spectra

Gutzwiller’s semiclassical trace formula for the density of states in a chaotic system diverges near bifurcations of periodic orbits, where it must be replaced by a modified one with uniform approximations. It is well known that, when applying these approximations, complex predecessors of orbits created in the bifurcation (“ghost orbits”) can produce clear signatures in the semiclassical spectra. We demonstrate that these orbits themselves can undergo bifurcations, resulting in complex, non-generic bifurcation scenarios. We do so by studying an example taken from the Diamagnetic Kepler Problem. By application of normal form theory, we construct an analytic description of the complete bifurcation scenario, which is then used to calculate the pertinent uniform approximation. The ghost orbit bifurcation turns out to produce signatures in the semiclassical spectrum in much the same way as a bifurcation of real orbits would.