  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.
