Test of the Periodic Orbit Approximation inn - Disk

The scattering of a point particle in two dimensions from two (or three) equally sized (and spaced) circular hard disks is one of the simplest classically hyperbolic scattering problems. Because of this simplicity such systems are well suited for the study of the semiclassical periodic orbit approximation in the cycle expansion of the dynamical zeta function applied to a quantum mechanical scattering problem. Especially the predictions of the semiclassical cycle expansion for the quantum mechanical resonances can be tested in these n-disk systems. Whereas for high wave numbers the cycle expansion gives quite accurate results, there are systematic deviations for low wave numbers from the exact quantum mechanical values. The low-lying quantum mechanical resonance poles of the 2-and 3-disk problem are constructed and compared to the cycle expansion results. The characteristic determinant of the scattering matrix is expanded in terms of simple traces which in turn are related to the classical periodic orbits and possible creeping contributions. It will be shown that for large separations of the disks the correct resonance pole positions can be extracted just from the knowledge of the lowest traces whose semiclassical limit are the fundamental periodic orbits. Creeping orbit corrections are shown to be small.