An analysis of the classical closed orbits in the infinitely deep elongated ellipsoidat well is given with respect to their relevance to the gross-shell structure of the single-particle spectra in deformed nuclei. It is shown that the deformed-shape gross-shells responsible for the ground state deformations, as well as those responsible for the fission intermediate states, find a reasonable explanation in terms of the semiclassical three-dimensional quantization involving the planar orbits in the axis-of-symmetry plane and the simplest three-dimensional orbits which appear at large distortions. The former are found important for the ground state shapes. The results are also compared with the harmonic oscillator model.

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 fargoing 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).

The isoscalar and isovector particle densities in the effective surface approximation to the average binding energy are used to derive analytical expressions of the surface symmetry energy, the neutron skin thickness and the isovector stiffness of sharp edged proton-neutron asymmetric nuclei. For most Skyrme forces the isovector coefficients of the surface energy and of the stiffness are significantly different from the empirical values derived in the liquid drop model. Using the analytical isovector surface energy constants in the framework of the hydrodynamical and the Fermiliquid droplet models the mean energies and the sum rules of the isovector giant dipole resonances are found to be in fair agreement with the experimental data.

Density distribution across the nuclear surface is obtained in the approximation of relatively sharp nuclear edge. It is used to determine dynamical parts of the density relevant to density vibration resonances. Results of the simple calculations are in close agreement with detailed microscopic theories.

In heavy nuclei where the thickness of the diffused edge is relatively small, a certain sharp effective surface can be defined which characterizes the shape of the nucleus, and it can be considered as a collective dynamic variable. It is shown that the problem of fluid dynamics can be simplified by reducing it to simple linearized equations for the dynamics in the nuclear interior and boundary conditions set at the effective dynamic sharp surface of the density distribution. These conditions are derived from the fluid dynamical equations. Transitional densities obtained from this simple model are compared with the numerical solution of fluid dynamical equations.

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

The quasiparticle Landau Fermi-liquid and periodic orbit theories are presented for the semiclassical description of collective excitations in nuclei, which are close to one of the main topics of the fruitful activity of S. T. Belyaev. Density-density response functions are studied at low temperatures within the temperature-dependent collisional Fermi-liquid theory in the relaxation time approximation. The isothermal, isolated (static) and adiabatic susceptibilities for nuclear matter show the ergodicity property. Temperature corrections to the response function, viscosity and thermal conductivity coefficients have been derived, also in the long wave-length (hydrodynamic) limit. The relaxation and correlation functions are obtained through the fluctuation-dissipation theorem and their properties are discussed in connection to the static susceptibilities and ergodicity of the Fermi systems. Transport coefficients, such as nuclear friction and inertia as functions of the temperature for the hydrodynamic (heat-pole and first sound) and Fermi-surface-distortion zero sound modes are derived within the Fermi-liquid droplet model. They are shown to be in agreement with the semi-microscopical calculations based on the nuclear shell model (SM) for large temperatures. This kinetic approach is extended to the study of the neutron-proton correlations in asymmetric neutronrich nuclei. The surface symmetry binding-energy constants are presented as functions of the Skyrme force parameters in the approximation of a sharp edged proton-neutron asymmetric nucleus and applied to calculations of the isovector giant dipole resonance. The energies, sum rules and transition densities of these resonances obtained by using analytical expression for these surface constants in terms of the Skyrme force parameters are in fairly good agreement with the experimental data. An analysis of the experimental data, in particular the specific structure of these resonances in terms of a main, and some satellite peaks, in comparison with our analytical approach and other theoretical semi-microscopical models, might turn out to be of capital importance for a better understanding of the values of the fundamental surface symmetry-energy constant. The semiclassical collective moment of inertia is derived analytically beyond the quantum perturbation approximation of the cranking model for any potential well as a mean field. It is shown that this moment of inertia can be approximated by its rigid-body value for the rotation with a given frequency within the ETF and more general periodic orbit theories in the nearly local long-length approximation. Its semiclassical shell-structure components are derived in terms of the periodic-orbit free-energy shell corrections. An enhancement of the moment of inertia near the symmetry-breaking bifurcation deformations was found. We obtained good agreement between the semiclassical and quantum shell-structure components of the moment of inertia for several critical bifurcation deformations for the completely analytically ...

We derive a semiclassical trace formula for the level density of the three-dimensional spheroidal cavity. To overcome the divergences and discontinuities occurring at bifurcation points and in the spherical limit, the trace integrals over the action-angle variables are performed using an improved stationary phase method. The resulting semiclassical level density oscillations and shell energies are in good agreement with quantum-mechanical results. We find that the births of three-dimensional orbits through the bifurcations of planar orbits in the equatorial plane lead to considerable enhancement of shell effect for superdeformed shapes. * ) In this paper SSPM denotes the standard stationary phase method and its extension to continuous symmetries. 3)-5), 7)