/ 99 08 01 9 v 1 1 8 A ug 1 99 9 Uniform approximations for non - generic bifurcation scenarios
نویسندگان
چکیده
Gutzwiller's trace formula allows interpreting the density of states of a classically chaotic quantum system in terms of classical periodic orbits. It diverges when periodic orbits undergo bifurcations, and must be replaced with a uniform approximation in the vicinity of the bifurcations. As a characteristic feature, these approximations require the inclusion of complex " ghost orbits ". By studying an example taken from the Diamag-netic Kepler Problem, viz. the period-quadrupling of the balloon-orbit, we demonstrate that these ghost orbits themselves can undergo bifurca-tions, giving rise to non-generic complicated bifurcation scenarios. We extend classical normal form theory so as to yield analytic descriptions of both bifurcations of real orbits and ghost orbit bifurcations. We then show how the normal form serves to obtain a uniform approximation taking the ghost orbit bifurcation into account. We find that the ghost bifurcation produces signatures in the semiclassical spectrum in much the same way as a bifurcation of real orbits does.
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Gutzwiller's trace formula for the semiclassical density of states in a chaotic system diverges near bifurcations of periodic orbits, where it must be replaced with uniform approximations. It is well known that, when applying these approximations, complex predecessors of orbits created in the bifurcation (" ghost orbits ") can produce pronounced signatures in the semiclassical spectra in the vi...
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