Sum Rules for Periodic Orbit Theory

Several families of exact sum rules for stabilities of
periodic points of *d*-dimensional polynomial mappings are
derived and applied to 1-*d* polynomial
maps, 3-disk repellers,
the Hénon map, and the Farey map.
The associated Fredholm determinants are of particularly
simple polynomial form. The theory developed
suggests an alternative to the standard periodic orbit approach
to determining eigenspectra of evolution operators.

P. Cvitanovic, K. Hansen and (G. Vattay)

Classical and Quantum Mechanical Properties of Hydrogen in a Magnetic Field

We investigate classical and quantum mechanical properties in the system of hydrogen in a magnetic field. The properties of the system with mixed chaotic an stable dynamics are especially treated.

K.T. Hansen and (G. Tanner)

Geometric Orbits in the Power Spectra of Waves

We investigate the relationship between the power spectrum of a wave field and that of a spatially uncorrelated source exciting it. Temporal correlations between values of the field at different positions are also examined. It is found that interference effects can significantly alter the structure of the power spectrum, leading to oscillations in it, even when the power spectrum of the source is a smooth function of frequency. We derive a semiclassical approximation in which these oscillations are related to orbits of the geometrical limit of the wave system. We also derive a trace formula that approximates a spatial average of the wave power spectrum as a sum over periodic orbits. These calculations explain the structure of a measured power spectrum of the fluctuating height of a fluid surface, generated by the circular hydraulic jump, which provided the motivation for the study.

(S.C. Creagh) and P. Dimon

Fredholm Theory and the Semiclassical Approximation

The semiclassical approximation for chaotic systems usually produces a divergent sum over classical trajectories. The Fredholm method is a way of combining Bogomolny's T-operator with the old Fredholm theory for solving integral equation, to obtain a general scheme of resummation valid for various quantities, not necessarily the density of states. We have applied it to individual wavefunctions, to get a resummed formula permitting to predict the occurence of scars (enhancement of wavefunctions around classical periodic orbits). We apply it also to scattering systems, such as a kicked 1-d system or the three disk, and find that it improves considerably the convergence of the semiclassical series for the scattering amplitudes. We also compare the results with the exact quantum quantities.

B. Georgeot, (S. Fishman and R.E. Prange)

Ray-Splitting and Quantum Chaos

Ray-splitting is a phenomenon whereby a ray incident on a boundary splits into more than one ray travelling away from the boundary. This corresponds to the short-wavelength limit of wave systems where at some place (discontinuity of the potential, boundary...) a wave is decomposed into two other (for example, one transmitted and one reflected). We extend results and techniques from quantum chaos to this kind of systems, and test these extensions on a simple model. This should enable to treat various kind of physical problems, in particular in accoustics.

B. Georgeot, (T.M. Antonsen, Jr., R. Bluemel, E. Ott and R.E. Prange)

Bogomolny's Transfer Operator

Bogomolny's transfer operator is still a new tool in the field of semiclassical quantization of chaotic systems. One study the properties of this operator (unitarity, size...) in different bases and for different systems, such as the three-disks systems, a kicked potential and the rectangular billiard.

B. Georgeot

Coulomb Blockade Conductance Peak Distributions in Quantum Dots and the Crossover between Orthogonal and Unitary Symmetries

Closed expressions are derived for the resonance widths and Coulomb blockade conductance peak heights in quantum dots for the crossover regime between conserved and broken time-reversal symmetry. The results hold for leads with any number of possibly correlated and inequivalent channels. Our analytic predictions are in good agreement with simulations of both random matrices and a chaotic billiard with a magnetic flux line.

(Y. Alhassid), (J. Hormuzdiar) and N.D. Whelan

Small Disks and Semiclassical Resonances

We study the effect on quantum spectra of the existence of small circular disks in a billiard system. In the limit where the disk radii vanish there is no effect, however this limit is approached very slowly so that even very small radii have comparatively large effects. We include diffractive orbits which scatter off the small disks in the periodic orbit expansion. This situation is formally similar to edge diffraction except that the disk radii introduce length scale in the problem such that for wave lengths smaller than the disk radius we recover the usual semi-classical approximation; however, for wave lengths larger than the disk radius there is a qualitatively different behaviour. We test the theory by successfully estimating the positions of scattering resonances in geometries consisting of three and four small disks.

P.E. Rosenqvist, N.D. Whelan and (A. Wirzba)

Fri Feb 21 15:17:28 MET 1997