Semiclassical catastrophe theory of simple bifurcations.

The Fedoriuk-Maslov catastrophe theory of caustics and turning points is extended to solve the bifurcation problems by the improved stationary phase method (ISPM). The trace formulas for the radial power-law (RPL) potentials are presented by the ISPM based on the second- and third-order expansion of the classical action near the stationary point. A considerable enhancement of contributions of the two orbits (pair consisting of the parent and newborn orbits) at their bifurcation is shown. The ISPM trace formula is proposed for a simple bifurcation scenario of Hamiltonian systems with continuous symmetries, where the contributions of the bifurcating parent orbits vanish upon approaching the bifurcation point due to the reduction of the end-point manifold. This occurs since the contribution of the parent orbits is included in the term corresponding to the family of the newborn daughter orbits. Taking this feature into account, the ISPM level densities calculated for the RPL potential model are shown to be in good agreement with the quantum results at the bifurcations and asymptotically far from the bifurcation points.

Ultrasonic waves in classical gases.

The velocity and absorption coefficient for the plane sound waves in a classical gas are obtained by solving the Boltzmann kinetic equation, which describes the reaction of the single-particle distribution function to a periodic external field. Within the linear response theory, the nonperturbative dispersion equation valid for all sound frequencies is derived and solved numerically. The results are in agreement with the approximate analytical solutions found for both the frequent- and rare-collision regimes. These results are also in qualitative agreement with the experimental data for ultrasonic waves in dilute gases.

Viscosity of a classical gas: The rare-collision versus the frequent-collision regime.

The shear viscosity η for a dilute classical gas of hard-sphere particles is calculated by solving the Boltzmann kinetic equation in terms of the weakly absorbed plane waves. For the rare-collision regime, the viscosity η as a function of the equilibrium gas parameters-temperature T, particle number density n, particle mass m, and hard-core particle diameter d-is quite different from that of the frequent-collision regime, e.g., from the well-known result of Chapman and Enskog. An important property of the rare-collision regime is the dependence of η on the external ("nonequilibrium") parameter ω, frequency of the sound plane wave, that is absent in the frequent-collision regime at leading order of the corresponding perturbation expansion. A transition from the frequent to the rare-collision regime takes place when the dimensionless parameter nd^{2}(T/m)^{1/2}ω^{-1} goes to zero. The scaled absorption coefficient for sound waves calculated in the rare and frequent-collision regimes is found to be in qualitative agreement with the experimental data.

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.

Semiclassical trace formula for the two-dimensional radial power-law potentials.

The trace formula for the density of single-particle levels in the two-dimensional radial power-law potentials, which nicely approximate up to a constant shift the radial dependence of the Woods-Saxon potential and its quantum spectra in a bound region, was derived by the improved stationary phase method. The specific analytical results are obtained for the powers α=4 and 6. The enhancement of periodic-orbit contribution to the level density near the bifurcations are found to be significant for the description of the fine shell structure. The semiclassical trace formulas for the shell corrections to the level density and the energy of many-fermion systems reproduce the quantum results with good accuracy through all the bifurcation (symmetry breaking) catastrophe points, where the standard stationary-phase method breaks down. Various limits (including the harmonic oscillator and the spherical billiard) are obtained from the same analytical trace formula.