We have derived an analytical trace formula for the level density of the Henon-Heiles potential using the improved stationary phase method, based on extensions of Gutzwillers semiclassical path integral approach. This trace formula has the correct li
mit to the standard Gutzwiller trace formula for the isolated periodic orbits far from all (critical) symmetry-breaking points. It continuously joins all critical points at which an enhancement of the semiclassical amplitudes occurs. We found a good agreement between the semi- classical and the quantum oscillating level densities for the gross shell structures and for the energy shell corrections, solving the symmetry breaking problem at small energies.
We propose a phenomenological approach for the equation of state of a unitary Fermi gas. The universal equation of state is parametrised in terms of Fermi-Dirac integrals. This reproduces the experimental data over the accessible range of fugacity an
d normalised temperature, but cannot describe the superfluid phase transition found in the MIT experiment cite{ku}. The most sensitive data for compressibility and specific heat at phase transition can, however, befitted by introducing into the grand partition function a pair of complex conjugate zeros lying in the complex fugacity plane slightly off the real axis.
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 primitiv
e 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 Langers 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 $n$.
We first give an overview of the shell-correction method which was developed by V. M. Strutinsky as a practicable and efficient approximation to the general selfconsistent theory of finite fermion systems suggested by A. B. Migdal and collaborators.
Then we present in more detail a semiclassical theory of shell effects, also developed by Strutinsky following original ideas of M. Gutzwiller. We emphasize, in particular, the influence of orbit bifurcations on shell structure. We first give a short overview of semiclassical trace formulae, which connect the shell oscillations of a quantum system with a sum over periodic orbits of the corresponding classical system, in what is usually called the periodic orbit theory. We then present a case study in which the gross features of a typical double-humped nuclear fission barrier, including the effects of mass asymmetry, can be obtained in terms of the shortest periodic orbits of a cavity model with realistic deformations relevant for nuclear fission. Next we investigate shell structures in a spheroidal cavity model which is integrable and allows for far-going analytical computation. We show, in particular, how period-doubling bifurcations are closely connected to the existence of the so-called superdeformed energy minimum which corresponds to the fission isomer of actinide nuclei. Finally, we present a general class of radial power-law potentials which approximate well the shape of a Woods-Saxon potential in the bound region, give analytical trace formulae for it and discuss various limits (including the harmonic oscillator and the spherical box potentials).
This paper is dedicated to the memory of Vilen Mitrofanovich Strutinsky who would have been 80 this year. His achievements in theoretical nuclear physics are briefly summarized. I discuss in more detail the most successful and far-reaching of them, n
amely (1) the shell-correction method and (2) the extension of Gutzwillers semiclassical theory of shell structure and its application to finite fermionic systems, and mention some applications in other domains of physics.
We investigate the particle and kinetic-energy densities for a system of $N$ fermions bound in a local (mean-field) potential $V(bfr)$. We generalize a recently developed semiclassical theory [J. Roccia and M. Brack, Phys. Rev. Lett. {bf 100}, 200408
(2008)], in which the densities are calculated in terms of the closed orbits of the corresponding classical system, to $D>1$ dimensions. We regularize the semiclassical results $(i)$ for the U(1) symmetry breaking occurring for spherical systems at $r=0$ and $(ii)$ near the classical turning points where the Friedel oscillations are predominant and well reproduced by the shortest orbit going from $r$ to the closest turning point and back. For systems with spherical symmetry, we show that there exist two types of oscillations which can be attributed to radial and non-radial orbits, respectively. The semiclassical theory is tested against exact quantum-mechanical calculations for a variety of model potentials. We find a very good overall numerical agreement between semiclassical and exact numerical densities even for moderate particle numbers $N$. Using a local virial theorem, shown to be valid (except for a small region around the classical turning points) for arbitrary local potentials, we can prove that the Thomas-Fermi functional $tau_{text{TF}}[rho]$ reproduces the oscillations in the quantum-mechanical densities to first order in the oscillating parts.
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 no
n-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.
We investigate the particle and kinetic-energy densities for $N$ non-interacting fermions confined in a local potential. Using Gutzwillers semi-classical Green function, we describe the oscillating parts of the densities in terms of closed non-period
ic classical orbits. We derive universal relations between the oscillating parts of the densities for potentials with spherical symmetry in arbitrary dimensions, and a ``local virial theorem valid also for arbitrary non-integrable potentials. We give simple analytical formulae for the density oscillations in a one-dimensional potential.
We report on transcritical bifurcations of periodic orbits in non-integrable two-dimensional Hamiltonian systems. We discuss their existence criteria and some of their properties using a recent mathematical description of transcritical bifurcations i
n families of symplectic maps. We then present numerical examples of transcritical bifurcations in a class of generalized Henon-Heiles Hamiltonians and illustrate their stabilities and unfoldings under various perturbations of the Hamiltonians. We demonstrate that for Hamiltonians containing straight-line librating orbits, the transcritical bifurcation of these orbits is the typical case which occurs also in the absence of any discrete symmetries, while their isochronous pitchfork bifurcation is an exception. We determine the normal forms of both types of bifurcations and derive the uniform approximation required to include transcritically bifurcating orbits in the semiclassical trace formula for the density of states of the quantum Hamiltonian. We compute the coarse-grained density of states in a specific example both semiclassically and quantum mechanically and find excellent agreement of the results.
We first give a short review of the ``local-current approximation (LCA), derived from a general variation principle, which serves as a semiclassical description of strongly collective excitations in finite fermion systems starting from their quantum-
mechanical mean-field ground state. We illustrate it for the example of coupled translational and compressional dipole excitations in metal clusters. We then discuss collective electronic dipole excitations in C$_{60}$ molecules (Buckminster fullerenes). We show that the coupling of the pure translational mode (``surface plasmon) with compressional volume modes in the semiclasscial LCA yields semi-quantitative agreement with microscopic time-dependent density functional (TDLDA) calculations, while both theories yield qualitative agreement with the recent experimental observation of a ``volume plasmon.
