Effect of pitchfork bifurcations on the spectral statistics of Hamiltonian systems

Abstract

We present a quantitative semiclassical treatment of the effects of bifurcations on the spectral rigidity and the spectral form factor of a Hamiltonian quantum system defined by two coupled quartic oscillators, which on the classical level exhibits mixed phase space dynamics. We show that the signature of a pitchfork bifurcation is two-fold: Beside the known effect of an enhanced periodic orbit contribution due to its peculiar $\hslash$-dependence at the bifurcation, we demonstrate that the orbit pair born {\em at} the bifurcation gives rise to distinct deviations from universality slightly {\em above} the bifurcation. This requires a semiclassical treatment beyond the so-called diagonal approximation. Our semiclassical predictions for both the coarse-grained density of states and the spectral rigidity, are in excellent agreement with corresponding quantum-mechanical results.

LaTex, 25 pp., 14 Figures (26 *.eps files); final version 3, to be published in Journal of Physics A

References in corpus (4)

• Semiclassical Foundation of Universality in Quantum Chaos
• Monodromy in the resonant swing spring
• Semiclassical form factor for spectral and matrix element fluctuations of multi-dimensional chaotic systems
• Classical basis for quantum spectral fluctuations in hyperbolic systems

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• Quantum Mechanical Approach to Bifurcation Point Detection in Hamiltonian Dynamical Systems