Geometric and diffractive orbits in the scattering from confocal hyperbolas

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Abstract
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.
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