Semiclassical Approach for Bifurcations in a Smooth Finite-Depth Potential

Abstract

The analytical trace formula for a dense cascade of bifurcations was derived using the improved stationary phase method based on extensions of the semiclassical Gutzwiller path integral approach. For the integrable version of the famous Hénon-Heiles Hamiltonian, our analytical trace formula solves all bifurcation problems, in particular, in the harmonic oscillator limit and the potential barrier limit. We obtain nice agreement with quantum results for gross to finer shell structures in level densities and for the shell structure energies, even near the potential barrier where there is a rather dense sequence of bifurcations. §1. IntroductionThe Gutzwiller trace formula 1), 2) and its extensions to continuous symmetries 3)-9) are nice tools to study shell structures in finite fermionic systems. This is the socalled periodic orbit theory (POT), which relates quantum fluctuations in singleparticle level densities to classical periodic orbits through their dynamical characteristics, such as action integrals, stability matrices, and degeneracy (symmetry) parameters.Recently, the POT was employed in studies seeking to overcome some symmetrybreaking problems related to divergencies and discontinuities of the standard stationary phase method (standard SPM, SSPM) 2), 9) due to a bifurcation phenomenon. For instance, the improved stationary phase method (ISPM) within the extended Gutzwiller approach (EGA) 10)-12) was used in derivations of the trace formulas. The ISPM is based on the theory of critical caustics and turning points formulated by Maslov and Fedoryuk 13)-17) in order to overcome bifurcation problems. Furthermore, the idea of Berry and Tabor 5) has been applied to calculate catastrophe integrals more exactly within finite limits over the accessible phase space volume for classical motion.Other semiclassical approaches, known as the uniform approximations, were suggested and successfully developed previously 18)-26) on the basis of the theory of

Shell structure and orbit bifurcations in finite fermion systems

“…The level density g(E) can be obtained from the semiclassical Green's function by taking the imaginary part of its trace in phase space variables, see [21], also the references therein,…”

“…The Maslov phase µ CT is determined by the number of caustic and turning points within the catastrophe theory by Fedoryuk and Maslov [21,22,23,24]. In (3),…”

“…12 For the following it is useful to note the classical degeneracies K of the above-mentioned POs [15,18,20,21]. The meridional (planar) orbits (denoted by "2D" henceforth for both elliptic 2DE…”

“…where b po = 1 for all K = 1 polygon-like PO families, except for the diameters for which one has (see [21])…”

“…2 are the predictions of the POT for the loci of the ground-state minima, using the shortest POs in a spheroidal cavity. Bifurcations of POs under the variation of a deformation parameter or the (Fermi) energy can have noticeable effects for the shell structure [5,6,7,18,19,20,21].…”