Uniform trace formulae forSU(2) andSO(3) symmetry breaking

Abstract

We develop uniform approximations for the trace formula for non-integrable systems in which SU(2) symmetry is broken by a non-linear term of the Hamiltonian. As specific examples, we investigate Hénon-Heiles type potentials. Our formalism can also be applied to the breaking of SO(3) symmetry in a three-dimensional cavity with axially-symmetric quadrupole deformation.PACS number: 03.65.SqNovember 5, 1998 J. Phys. A, in print * ) Present address:

Semiclassical trace formulas for pitchfork bifurcation sequences

“…(The same limit was obtained in a uniform approximation for the full non-integrable Hénon-Heiles potential in [12] neglecting, however, the bifurcations.) In the figures 14 and 15, we compare the results obtained from the grand uniform approximation (49) with those of quantum-mechanical calculations for system (38) with λ = 0.04 (with saddle energy E * = 104.666 corresponding to e = 1), both coarse-grained by a Gaussian convolution with an energy range γ = 0.1, including repetition numbers up to |k u |, |k v | ≤ 8 into the semiclassical trace formula (49).…”

“…In [12], a uniform approximation for the symmetry breaking at e = 0 was developed which continuously interpolates from the harmonicoscillator limit, given in (50) below, to the region where the Gutzwiller trace formula for the isolated orbits is valid. However, the bifurcations of the A orbit have not been treated uniformly in the references [12,35], so that the accuracy of the results decreased near the saddle at e = 1. In [18] the classical bifurcation cascade in the Hénon-Heiles potential was discussed, in which the sequence of two successive pitchfork bifurcations repeats itself infinitely often.…”

“…The chaotic regions increase through the destruction of rational tori when continuous symmetries are broken, and through bifurcations of periodic orbits when the energy or another control parameter of the system is increased. Explicit semiclassical trace formulae have been given for various systems with continous symmetries [3,5,6,7], for symmetry breaking through the destruction of rational tori [8,9,10,11,12], and for isolated bifurcations [13,14,15]. However more complicated bifurcation scenarios which usually occur in realistic physical systems and, in particular, bifurcation cascades [16,17,18,19] still constitute one of the most serious problems of the semiclassical theory.…”

Periodic-orbit bifurcations and superdeformed shell structure

“…At these points the stationary-phase approximation, used in the semiclassical evaluation of the trace integrals, breaks down. Various ways of avoiding these divergences have been studied [2,4,16], some of them employing uniform approximations [17][18][19][20]. Here we employ an improved stationary-phase method (ISPM) for the evaluation of the trace integrals in the phase-space representation, based on the studies in Refs.…”

Periodic orbit theory for the Hénon-Heiles system in the continuum region

“…For semiclassical calculations at e < 1, we refer to earlier papers [18,28]. The periodic orbits in the region e > 1 are not all unstable.…”