HOW CHAOTIC IS THE STADIUM BILLIARD - A SEMICLASSICAL ANALYSIS

The impression gained from the literature published to date is that th e spectrum of the stadium billiard can be adequately described, semicl assically, by the Gutzwiller periodic orbit trace formula together wit h a modified treatment of the marginally stable family of bouncing-bal l orbits. I show that this belief is erroneous. The Gutzwiller trace f ormula is not applicable for the phase-space dynamics near the bouncin g-ball orbits. Unstable periodic orbits close to the marginally stable family in phase space cannot be treated as isolated stationary phase points when approximating the trace of the Green's function. Semiclass ical contributions to the trace show an A-dependent transition from ha rd chaos to integrable behaviour for trajectories approaching the boun cing-ball orbits. A whole region in phase space surrounding the margin al stable family acts, semiclassically, like a stable island with boun daries being explicitly A-dependent. The localized bouncing-ball state s found in the billiard derive from this semiclassically stable island . The bouncing-ball orbits themselves, however, do not contribute to i ndividual eigenvalues in the spectrum. An EBK-like quantization of the regular bouncing-ball eigenstates in the stadium can be derived. The stadium billiard is thus an ideal model for studying the influence of almost regular dynamics near marginally stable boundaries on quantum m echanics. The behaviour is generically found at the border of classica lly stable islands in systems with a mixed phase-space structure.