Semiclassical trace formulas for pitchfork bifurcation sequences

In nonintegrable 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 quantum-mechanical 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.

Figure 1

Contour plots of the normal form (8) in dependence of the parameters ${ϵ}_i$ for the case $a=-1$. From left to right: ${ϵ}_2<{ϵ}_1<0$, ${ϵ}_2<0<{ϵ}_1$, and $0<{ϵ}_2<{ϵ}_1$.

Figure 2

Scaled double-well potential. Left: contour plot with the four shortest periodic orbits $A$ and $B$ evaluated at $e=0.96$. Right: cut of the potential along $u=0$.

Figure 3

Orbits participating in a pitchfork bifurcation sequence in the double-well potential (21). Upper row: real part (left) and imaginary part (middle) of ghost orbit $R$ at $e=\mathrm{0.908\hspace{0.17em}64}$, and real orbit $R$ at $e=0.95$ (right). Lower row: real part (left) and imaginary part (middle) of ghost orbit $L$ at $e=0.94$, and real orbit $L$ at $e=0.95$ (right).

Figure 4

Properties of the periodic orbits $A$, $R$, and $L$ near their bifurcations in the double-well potential (21), plotted versus the scaled energy $e$. Top left, stability traces; middle left, action differences; bottom left, periods; and right, Gutzwiller amplitudes (cf. text). The dashed portions of all curves correspond to the complex pre-bifurcation ghost orbits.

Figure 5

Oscillating part of density of states in the double-well potential (21). Solid line: exact quantum result obtained with $\lambda =0.0008$. Dashed line: uniform approximation including isolated contribution of orbit $B$. Dotted line: sum of Gutzwiller contributions of isolated orbits, diverging at the two lowest bifurcations of the $A$ orbit. (The other bifurcations, lying at $e>0.9998$, cannot be seen at this resolution.) Coarse graining by Gaussian convolution with energy width $\gamma =0.5$.

Figure 6

The Hénon-Heiles potential. Left: equipotential contour lines in scaled energy units $e$ in the plane of scaled variables $u,v$. The dashed lines are the symmetry axes. The three shortest periodic orbits $A$, $B$, and $C$ (evaluated at the energy $e=1$) are shown by the heavy solid lines. Right: cut of the scaled potential along $u=0$.

Figure 7

Orbits born in the first pitchfork bifurcation sequence in the Hénon-Heiles potential. Upper row: real part (left) and imaginary part (middle) of ghost orbit $R_5$ at $e=0.9690$, and real orbit $R_5$ at $e=0.9798$ (right). Lower row: real part (left) and imaginary part (middle) of ghost orbit $L_6$ at $e=0.9864$, and real orbit $L_6$ at $e=0.9870$ (right).

Figure 8

Poincaré surfaces of section (PSS) of the scaled Hénon-Heiles Hamiltonian (28), taken for $v=0$. Left, $e=0.969$; middle, $e=0.982$; right, $e=0.989$.

Figure 9

The same as Fig. 4 for the Hénon-Heiles potential near the first two bifuractions of the $A$ orbit.

Figure 10

Oscillating part of density of states in the Hénon-Heiles potential. Solid lines: quantum-mechanical results obtained for $\lambda =0.03$. Dotted lines: sum of Gutzwiller contributions (6) of all isolated orbits. Dashed lines: codimension-two uniform approximation (11) for the orbits $A$, $R_5$, and $L_6$, including orbits $C$ and $D$ in the codimension-1 uniform approximation of Ref. 14 and the isolated $B$ orbit. Coarse graining with Gaussian width $\gamma =0.4$.

Figure 11

Equipotential lines in the ($u$,$v$) plane for the separable version of the Hénon-Heiles potential. The heavy solid lines show the two shortest periodic orbits $A$ and $B$ evaluated at $e=1$.

Figure 12

The same as Fig. 4 for the separable Hénon-Heiles system (37) for the bifurcation of the $k_u:k_v=5:3$ resonance at energy $e=\mathrm{0.987\hspace{0.17em}655}$. The central $A$ orbit is labeled by “0,” the bifurcated 5:3 torus by “1.”

Figure 13

Oscillating part of level density for the separable Hénon-Heiles system (37) near the 5:3 resonance, coarse-grained with a Gaussian width $\gamma =0.1$. Solid line, quantum-mechanical result obtained with $\lambda =0.04$; dotted line, sum of Berry-Tabor contribution of 5:3 torus and Gutzwiller contribution of isolated $A$ orbit; dashed line, uniform approximation (18).

Figure 14

Oscillating part of level density of the separable Hénon-Heiles system (37), coarse grained with $\gamma =0.1$. Solid lines: quantum-mechanical result for $\lambda =0.04$. Dashed lines: semiclassical results with $k_u,k_v⩽8$.

Figure 15

Oscillating part of level density of the separable Hénon-Heiles system (37), coarse grained with $\gamma =0.1$. Solid lines: quantum-mechanical result. Dashed lines: semiclassical results with $k_u,k_v⩽8$. Top: Berry-Tabor result for the tori. Center: sum of Berry-Tabor result for the tori plus Gutzwiller result for the isolated $A$ orbit. Bottom: uniform approximation (49).

Figure 16

Shell structure in the level density of the integrable Hénon-Heiles system. Solid lines: exact EBK integral (C15). Dashed lines: asymptotic approximations. Upper panel: Berry-Tabor contributions (C18) of the $T$ tori plus Gutzwiller contribution (C23) of the isolated $A$ orbit. Lower panel: uniform approximation (49). In both cases, repetition numbers $\mid k_u,k_v\mid ⩽8$ are included. Here the lowest energy region is shown where the isolated $A$ orbit contribution diverges in the limit $e\to 0$.

Figure 17

The same as Fig. 16 in an intermediate energy region. The top panel exhibits the divergence of the Gutzwiller contributions of the $A$ orbit (dashed line) near the $k_u:k_v=9:8$ and 8:7 resonances.

Figure 18

The same as Fig. 16 in the top energy range, covering the resonances with $k_u:k_v⩾3:2$.