Geometric and Diffractive Orbits in the Scattering from Confocal Hyperbolae

We study the scattering resonances between two confocal hyperbolae 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 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.