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

Abstract

We discuss the coarse-grained level density of the Hénon-Heiles system above the barrier energy, where the system is nearly chaotic. We use periodic orbit theory to approximate its oscillating part semiclassically via Gutzwiller's semiclassical trace formula (extended by uniform approximations for the contributions of bifurcating orbits). Including only a few stable and unstable orbits, we reproduce the quantum-mechanical density of states very accurately. We also present a perturbative calculation of the stabilities of two infinite series of orbits (R n and L m ), emanating from the shortest librating straight-line orbit (A) in a bifurcation cascade just below the barrier, which at the barrier have two common asymptotic Lyapunov exponents χ R and χ L

Semiclassical treatment of symmetry breaking and bifurcations in a non-integrable potential

“…As shown in [21,22], the trace T rM A is in this approximation in good agreement with the numerical results [20] near the saddle e → 1.…”

“…with the diagonal matrix elements given in (A9). For comparison, we recall the solution for the trace T rM A near the saddle e → 1 obtained in [21,22] in terms of the Legendre functions by using in (A2) the approximation of the Jacobi elliptic function, sn(z, k) ≈ tanh(z), i.e., by the zero-order term of its expansion near the saddle in powers of 1 − k 2 [29]: [20,9]); heavy dots show the approximation (A10) through the Mathieu functions (A9); the dashed line shows the improved Legendre function approximation with the constant (A12). Light dots present the asymptotic Legendre function approximation with r = 1 (rcorr = 0).…”

“…For comparison, we recall the solution for the trace T rM A near the saddle e → 1 obtained in [21,22] in terms of the Legendre functions by using in (A2) the approximation of the Jacobi elliptic function, sn(z, k) ≈ tanh(z), i.e., by the zero-order term of its expansion near the saddle in powers of 1 − k 2 [29]: Fig. 6: Numerical and analytical traces T rMA for A orbit.…”

“…where M PO is the stability matrix for the PO. Numerical and analytical calculations and the remarkable "fan" structure of the pitchfork bifurcations of the straight-line orbits A σ were analyzed in the case of the HH potential [20,21,22]. As shown in the Appendix (Fig.…”

Shells, orbit bifurcations, and symmetry restorations in Fermi systems

“…As shown in [85,93], the trace Tr M A is in this approximation in good agreement with the numerical results [81] near the saddle e → 1.…”