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.

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

Sewing sound quantum flesh onto classical bones

Semiclassical transformation theory implies an integral representation for stationary-state wave functions $\psi_m(q)$ in terms of angle-action variables ($\theta,J$). It is a particular solution of Schr\"{o}dinger's time-independent equation when terms of order $\hbar^2$ and higher are omitted, but the pre-exponential factor $A(q,\theta)$ in the integrand of this integral representation does not possess the correct dependence on $q$. The origin of the problem is identified: the standard unitarity condition invoked in semiclassical transformation theory does not fix adequately in $A(q,\theta)$ a factor which is a function of the action $J$ written in terms of $q$ and $\theta$. A prescription for an improved choice of this factor, based on succesfully reproducing the leading behaviour of wave functions in the vicinity of potential minima, is outlined. Exact evaluation of the modified integral representation via the Residue Theorem is possible. It yields wave functions which are not, in general, orthogonal. However, closed-form results obtained after Gram-Schmidt orthogonalization bear a striking resemblance to the exact analytical expressions for the stationary-state wave functions of the various potential models considered (namely, a P\"{o}schl-Teller oscillator and the Morse oscillator).Comment: RevTeX4, 6 page

Sum Rule Approach to the Isoscalar Giant Monopole Resonance in Drip Line Nuclei

Using the density-dependent Hartree-Fock approximation and Skyrme forces together with the scaling method and constrained Hartree-Fock calculations, we obtain the average energies of the isoscalar giant monopole resonance. The calculations are done along several isotopic chains from the proton to the neutron drip lines. It is found that while approaching the neutron drip line, the scaled and the constrained energies decrease and the resonance width increases. Similar but smaller effects arise near the proton drip line, although only for the lighter isotopic chains. A qualitatively good agreement is found between our sum rule description and the presently existing random phase approximation results. The ability of the semiclassical approximations of the Thomas-Fermi type, which properly describe the average energy of the isoscalar giant monopole resonance for stable nuclei, to predict average properties for nuclei near the drip lines is also analyzed. We show that when hbar corrections are included, the semiclassical estimates reproduce, on average, the quantal excitation energies of the giant monopole resonance for nuclei with extreme isospin values.Comment: 31 pages, 12 figures, revtex4; some changes in text and figure

Simple Analytical Particle and Kinetic Energy Densities for a Dilute Fermionic Gas in a d-Dimensional Harmonic Trap

We derive simple analytical expressions for the particle density $\rho(r)$ and the kinetic energy density $\tau(r)$ for a system of noninteracting fermions in a $d-$dimensional isotropic harmonic oscillator potential. We test the Thomas-Fermi (TF, or local-density) approximation for the functional relation $\tau[\rho]$ using the exact $\rho(r)$ and show that it locally reproduces the exact kinetic energy density $\tau(r)$, {\it including the shell oscillations,} surprisingly well everywhere except near the classical turning point. For the special case of two dimensions (2D), we obtain the unexpected analytical result that the integral of $\tau_{TF}[\rho(r)]$ yields the {\it exact} total kinetic energy.Comment: 4 pages, 4 figures; corrected versio

Analytical perturbative approach to periodic orbits in the homogeneous quartic oscillator potential

We present an analytical calculation of periodic orbits in the homogeneous quartic oscillator potential. Exploiting the properties of the periodic Lam{\'e} functions that describe the orbits bifurcated from the fundamental linear orbit in the vicinity of the bifurcation points, we use perturbation theory to obtain their evolution away from the bifurcation points. As an application, we derive an analytical semiclassical trace formula for the density of states in the separable case, using a uniform approximation for the pitchfork bifurcations occurring there, which allows for full semiclassical quantization. For the non-integrable situations, we show that the uniform contribution of the bifurcating period-one orbits to the coarse-grained density of states competes with that of the shortest isolated orbits, but decreases with increasing chaoticity parameter $\alpha$.Comment: 15 pages, LaTeX, 7 figures; revised and extended version, to appear in J. Phys. A final version 3; error in eq. (33) corrected and note added in prin

Occurrence of periodic Lam\'e functions at bifurcations in chaotic Hamiltonian systems

We investigate cascades of isochronous pitchfork bifurcations of straight-line librating orbits in some two-dimensional Hamiltonian systems with mixed phase space. We show that the new bifurcated orbits, which are responsible for the onset of chaos, are given analytically by the periodic solutions of the Lam\'e equation as classified in 1940 by Ince. In Hamiltonians with C_${2v}$ symmetry, they occur alternatingly as Lam\'e functions of period 2K and 4K, respectively, where 4K is the period of the Jacobi elliptic function appearing in the Lam\'e equation. We also show that the two pairs of orbits created at period-doubling bifurcations of touch-and-go type are given by two different linear combinations of algebraic Lam\'e functions with period 8K.Comment: LaTeX2e, 22 pages, 14 figures. Version 3: final form of paper, accepted by J. Phys. A. Changes in Table 2; new reference [25]; name of bifurcations "touch-and-go" replaced by "island-chain

Dynamical screening of the Coulomb interaction for two confined electrons in a magnetic field

We show that a difference in time scales of vertical and lateral dynamics permits one to analyze the problem of interacting electrons confined in an axially symmetric three-dimensional potential with a lateral oscillator confinement by means of the effective two-dimensional Hamiltonian with a screened Coulomb interaction. Using an adiabatic approximation based on action-angle variables, we present solutions for the effective charge of the Coulomb interaction (screening) for a vertical confinement potential simulated by parabolic, square, and triangular wells. While for the parabolic potential the solution for the effective charge is given in a closed anlytical form, for the other cases similar solutions can be easily calculated numerically.Comment: 10 pages, 6 figure

Semiclassical theory of weak antilocalization and spin relaxation in ballistic quantum dots

We develop a semiclassical theory for spin-dependent quantum transport in ballistic quantum dots. The theory is based on the semiclassical Landauer formula, that we generalize to include spin-orbit and Zeeman interaction. Within this approach, the orbital degrees of freedom are treated semiclassically, while the spin dynamics is computed quantum mechanically. Employing this method, we calculate the quantum correction to the conductance in quantum dots with Rashba and Dresselhaus spin-orbit interaction. We find a strong sensitivity of the quantum correction to the underlying classical dynamics of the system. In particular, a suppression of weak antilocalization in integrable systems is observed. These results are attributed to the qualitatively different types of spin relaxation in integrable and chaotic quantum cavities.Comment: 20 page

Analytic approach to bifurcation cascades in a class of generalized H\'enon-Heiles potentials

We derive stability traces of bifurcating orbits in H\'enon-Heiles potentials near their saddlesComment: LaTeX revtex4, 38 pages, 7 PostScript figures, 2 table