Semiclassical trace formulas for pitchfork bifurcation sequences

Abstract

In non-integrable Hamiltonian systems with mixed phase space and discrete symmetries, sequences of 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 uniform approximation for the combined contribution of pitchfork bifurcation pairs. For a two-dimensional double-well potential and the familiar Hénon-Heiles potential, we obtain very good agreement 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 excellent agreement with the slightly coarse-grained quantum-mechanical density of states.

Level Density of the Hénon-Heiles System Above the Critical Barrier Energy

“…For semiclassical calculations of the HH level density for e < 1, we refer to earlier papers [6,7]. In [8] we have calculated the complex resonance energies E m − iΓ m by the standard method of complex rotation, diagonalizing (1) in a finite harmonic-oscillator basis. The level density is, after subtracting the non-resonant part of the continuum, given by…”

“…which can be done analytically [8]. Its oscillating part, which describes the grossshell structure in the quantum-mechanical level density, is then given by…”

“…where ∆g(E) is the smooth part of (2) which we have extracted by a complex version [8] of the numerical Strutinsky averaging procedure [9]. Semiclassically, the quantity δg(E) can be approximated by Gutzwiller's trace formula [10], which for a system with two degrees of freedom reads…”

Fractal Weyl law behavior in an open Hamiltonian system

“…Here we do not consider resonances directly associated with these islands. However, the Coriolis terms has the effect of bringing out the structure of the repeller more clearly than for the pure HH system (ω = 0 [17]). Thus, varying ω allows for the fine tuning of the dynamics in the energy regime of interest.…”

“…The method of complex rotation was used to compute the complex resonance energies E n = E r − iΓ n /2 where Γ n is the resonance width [19]. This was accomplished by rotating the coordinates into the complex plane by an angle θ, i.e., q i → q i e iθ and then diagonalizing the resulting Hamiltonian matrix in a two-dimensional isotropic oscillator basis |n, m [17]. In principle the procedure is straightforward although care must be exercised to ensure that resonances are distinguished from scattering states.…”