Uniform approximations for pitchfork bifurcation sequences

In Hamiltonian systems with mixed phase space and discrete symmetries, sequences of isochronous pitchfork bifurcations of periodic orbits pave the way from integrability to chaos. In extending the semiclassical trace formula for the spectral density, we develop a new codimension-two uniform approximation for the combined contribution of two successive pitchfork bifurcations. For a two-dimensional double-well potential and the familiar Hénon-Heiles potential, we obtain an essential improvement over existing semiclassical approaches when comparing with exact quantummechanical calculations. We also consider the integrable limit of the scenario which corresponds to the bifurcation of a torus from an isolated periodic orbit. For the separable version of the Hénon-Heiles system we give an analytical uniform trace formula, which also yields the correct harmonic-oscillator SU(2) limit at low energies, and obtain perfect agreement with the slightly coarse-grained quantum-mechanical density of states.