Validity of the semiclassical periodic orbit approximation in the two- and three-disk problems

The high-lying resonances in the quantum mechanical scattering problem of a point particle from two or three equally sized (and spaced) circular hard disks in the two-dimensional plane are predicted quite well by the classical cycle expansion. There are, however, noticeable deviations for the lowest resonances. Therefore, the leading corrections from creeping paths to the cycle expansion in the two-disk scattering problem are constructed. Generalizations to the three-disk problem are indicated. The size of the corrections are estimated. They are shown to be too small to account for the deviations mentioned above. Finally, arguments are given that, for the two- and three-disk problem, the semiclassical predictions of the low-lying resonance poles are bound to fail.