The Accuracy of Semiclassical Quantization for Integrable Systems

The eigenvalues of the Hyperspherical billiard are calculated in the semiclassical approximation. The eigenvalues where this approximation fails are identified and found to be related to caustics that approach the wall of the billiard. The fraction of energy levels for which the semiclassical error is larger than some given value is calculated analytically (and tested numerically) and found to be independent of energy. The implications for other systems, in particular integrable ones, are discussed.