Comment on "Recurrences without closed orbits"

In a recent paper Robicheaux and Shaw [Phys. Rev. A 58, 1043 (1998)] calculate the recurrence spectra of atoms in electric fields with non-vanishing angular momentum not equal to 0. Features are observed at scaled actions ``an order of magnitude shorter than for any classical closed orbit of this system.'' We investigate the transition from zero to nonzero angular momentum and demonstrate the existence of short closed orbits with L_z not equal to 0. The real and complex ``ghost'' orbits are created in bifurcations of the ``uphill'' and ``downhill'' orbit along the electric field axis, and can serve to interpret the observed features in the quantum recurrence spectra.Comment: 2 pages, 1 figure, REVTE

Exceptional points in the elliptical three-disk scatterer using semiclassical periodic orbit quantization

The three-disk scatterer has served as a paradigm for semiclassical periodic orbit quantization of classical chaotic systems using Gutzwiller's trace formula. It represents an open quantum system, thus leading to spectra of complex eigenenergies. An interesting general feature of open quantum systems described by non-Hermitian operators is the possible existence of exceptional points where not only the complex eigenvalues but also their respective eigenvectors coincide. Using Gutzwiller's periodic orbit theory we show that exceptional points exist in a three-disk scatterer if the system's geometry is modified by extending the system from circular to elliptical disks. The extension is implemented in such a way that the system's characteristic $C_{3\mathrm{v}}$ symmetry is preserved. The two-dimensional parameter plane of the system is then spanned by the distance between and the excentricity of the elliptical disks. As typical signatures of exceptional points we observe the permutation of two resonances when an exceptional point is encircled in parameter space, and a non-Lorentzian resonance line shape in the weighted density of states.Comment: 7 pages, 7 figures, 1 tabl

Periodic orbit quantization of chaotic systems with strong pruning

The three-disk system, which for many years has served as a paradigm for the usefulness of cycle expansion methods, represents an extremely hard problem to semiclassical quantization when the disks are moved closer and closer together, since (1) pruning of orbits sets in, rendering the symbolic code incomplete, and (2) the number of orbits necessary to obtain accurate semiclassical eigenvalues proliferates exponentially. In this note we show that an alternative method, viz. harmonic inversion, which does not rely on the existence of complete symbolic dynamics or other specific properties of systems, provides a key to solving the problem of semiclassical quantization of systems with strong pruning. For the closed three-disk system we demonstrate how harmonic inversion, augmented by a signal cross-correlation technique, allows one to semiclassically calculate the energies up to the 28th excited state with high accuracy.Comment: 9 pages, 3 figures, submitted to Phys. Lett.