Closed orbits and spatial density oscillations in the circular billiard

We present a case study for the semiclassical calculation of the oscillations in the particle and kinetic-energy densities for the two-dimensional circular billiard. For this system, we can give a complete classification of all closed periodic and non-periodic orbits. We discuss their bifurcations under variation of the starting point r and derive analytical expressions for their properties such as actions, stability determinants, momentum mismatches and Morse indices. We present semiclassical calculations of the spatial density oscillations using a recently developed closed-orbit theory [Roccia J and Brack M 2008 Phys. Rev. Lett. 100 200408], employing standard uniform approximations from perturbation and bifurcation theory, and test the convergence of the closed-orbit sum.Comment: LaTeX, 42 pp., 17 figures (24 *.eps files, 1 *.tex file); final version (v3) to be published in J. Phys.

Super-shell structure in harmonically trapped fermionic gases and its semi-classical interpretation

It was recently shown in self-consistent Hartree-Fock calculations that a harmonically trapped dilute gas of fermionic atoms with a repulsive two-body interaction exhibits a pronounced {\it super-shell} structure: the shell fillings due to the spherical harmonic trapping potential are modulated by a beat mode. This changes the ``magic numbers'' occurring between the beat nodes by half a period. The length and amplitude of the beating mode depends on the strength of the interaction. We give a qualitative interpretation of the beat structure in terms of a semiclassical trace formula that uniformly describes the symmetry breaking U(3) $\to$ SO(3) in a 3D harmonic oscillator potential perturbed by an anharmonic term $\propto r^4$ with arbitrary strength. We show that at low Fermi energies (or particle numbers), the beating gross-shell structure of this system is dominated solely by the two-fold degenerate circular and (diametrically) pendulating orbits.Comment: Final version of procedings for the 'Nilsson conference

Semiclassical description of shell effects in finite fermion systems

A short survey of the semiclassical periodic orbit theory, initiated by M. Gutzwiller and generalized by many other authors, is given. Via so-called semiclassical trace formmulae, gross-shell effects in bound fermion systems can be interpreted in terms of a few periodic orbits of the corresponding classical systems. In integrable systems, these are usually the shortest members of the most degenerate families or orbits, but in some systems also less degenerate orbits can determine the gross-shell structure. Applications to nuclei, metal clusters, semiconductor nanostructures, and trapped dilute atom gases are discussed.Comment: LaTeX (revteX4) 6 pages; invited talk at Int. Conference "Finite Fermionic Systems: Nilsson Model 50 Years", Lund, Sweden, June 14-18, 200

A semiclassical analysis of the Efimov energy spectrum in the unitary limit

We demonstrate that the (s-wave) geometric spectrum of the Efimov energy levels in the unitary limit is generated by the radial motion of a primitive periodic orbit (and its harmonics) of the corresponding classical system. The action of the primitive orbit depends logarithmically on the energy. It is shown to be consistent with an inverse-squared radial potential with a lower cut-off radius. The lowest-order WKB quantization, including the Langer correction, is shown to reproduce the geometric scaling of the energy spectrum. The (WKB) mean-squared radii of the Efimov states scale geometrically like the inverse of their energies. The WKB wavefunctions, regularized near the classical turning point by Langer's generalized connection formula, are practically indistinguishable from the exact wave functions even for the lowest ($n=0$) state, apart from a tiny shift of its zeros that remains constant for large $n$.Comment: LaTeX (revtex 4), 18pp., 4 Figs., already published in Phys. Rev. A but here a note with a new referece is added on p. 1

On the canonically invariant calculation of Maslov indices

After a short review of various ways to calculate the Maslov index appearing in semiclassical Gutzwiller type trace formulae, we discuss a coordinate-independent and canonically invariant formulation recently proposed by A Sugita (2000, 2001). We give explicit formulae for its ingredients and test them numerically for periodic orbits in several Hamiltonian systems with mixed dynamics. We demonstrate how the Maslov indices and their ingredients can be useful in the classification of periodic orbits in complicated bifurcation scenarios, for instance in a novel sequence of seven orbits born out of a tangent bifurcation in the H\'enon-Heiles system.Comment: LaTeX, 13 figures, 3 tables, submitted to J. Phys.

Semiclassical trace formulae for systems with spin-orbit interactions: successes and limitations of present approaches

We discuss the semiclassical approaches for describing systems with spin-orbit interactions by Littlejohn and Flynn (1991, 1992), Frisk and Guhr (1993), and by Bolte and Keppeler (1998, 1999). We use these methods to derive trace formulae for several two- and three-dimensional model systems, and exhibit their successes and limitations. We discuss, in particular, also the mode conversion problem that arises in the strong-coupling limit.Comment: LaTeX2e, 25 pages incl. 9 figures, version 3: final version in print for J. Phys.

Periodic-Orbit Bifurcations and Superdeformed Shell Structure

We have derived a semiclassical trace formula for the level density of the three-dimensional spheroidal cavity. To overcome the divergences occurring at bifurcations and in the spherical limit, the trace integrals over the action-angle variables were performed using an improved stationary phase method. The resulting semiclassical level density oscillations and shell-correction energies are in good agreement with quantum-mechanical results. We find that the bifurcations of some dominant short periodic orbits lead to an enhancement of the shell structure for "superdeformed" shapes related to those known from atomic nuclei.Comment: 4 pages including 3 figure

Curvature Correction in the Strutinsky's Method

Mass calculations carried out by Strutinsky's shell correction method are based on the notion of smooth single particle level density. The smoothing procedure is always performed using curvature correction. In the presence of curvature correction a smooth function remains unchanged if smoothing is applied. Two new curvature correction methods are introduced. The performance of the standard and new methods are investigated using harmonic oscillator and realistic potentials.Comment: 4 figures, submitted to Journal of Physics G: Nuclear and Particle Physic