Diffractive Orbits Research Articles

Edge diffraction, trace formulae and the cardioid billiard

We study the effect of edge diffraction on the semiclassical analysis of two-dimensional quantum systems by deriving a trace formula which incorporates paths hitting any number of vertices embedded in an arbitrary potential. This formula is used to study the cardioid billiard, which has a single vertex. The formula works well for most of the short orbits we analysed but fails for a few diffractive orbits due to a breakdown in the formalism for certain geometries. We extend the symbolic dynamics to account for diffractive orbits and use it to show that in the presence of parity symmetry the trace formula decomposes in an elegant manner such that for the cardioid billiard the diffractive orbits have no effect on the odd spectrum. Including diffractive orbits helps resolve peaks in the density of even states but does not appear to affect their positions. An analysis of the level statistics shows no significant difference between spectra with and without diffraction.

Semiclassical quantization using diffractive orbits.

Diffraction, in the context of semiclassical mechanics, describes the manner in which quantum mechanics smooths over discontinuities in classical mechanics. Semiclassical quantization then requires the inclusion of diffractive periodic orbits in addition to classical periodic orbits. In this Letter we construct the corresponding diffractive zeta function and apply it to a scattering problem which has only diffractive periodic orbits. We find that the resonances are accurately given by the zeros of the zeta function.

Diffractive Orbits in Quantum Billiards

Diffractive Orbits in Quantum Billiards

Diffractive orbits in quantum billiards.

We study diffractive effects in two dimensional polygonal billiards. We derive an analytical trace formula accounting for the role of non-classical diffractive orbits in the quantum spectrum. As an illustration the method is applied to a triangular billiard.

Geometric and diffractive orbits in the scattering from confocal hyperbolas.

We study the scattering resonances between two confocal hyperbolas and show that the spectrum is dominated by the effect of a single periodic orbit. There are two distinct cases, depending on whether the orbit is geometric or diffractive. A generalization of periodic orbit theory allows us to incorporate the second possibility. In both cases, we also perform a Wentzel-Kramers-Brillouin (WKB) analysis. Although it is found that the semiclassical approximations work best for resonances with large energies and narrow widths, there is reasonable agreement even for resonances with large widths---unlike the two disk scatterer. We also find agreement with the next order correction to periodic orbit theory.

Shell structure in faceted metal clusters.

We study the quantized electronic energy levels in a three-dimensional icosahedral billiard modeling a faceted metal cluster. The first 2000 levels are determined numerically. The magic numbers are compared with experimental data and with the results for a spherical model. We discuss the supershell structure and propose its study as a test of cluster sphericity. We compare our results with the predictions of the semiclassical trace formula and point out the relevance of diffractive orbits.