Penumbra diffraction in the quantization of concave billiards
  The semiclassical description of billiard spectra is extended to include the diffractive contributions from orbits which are nearly tangent to a concave part of the boundary. The leading correction for an unstable isolated orbit is of the same order as the standard Gutzwiller expression itself. The importance of the diffraction corrections is further emphasized by an estimate which shows that for any large fixed k almost all contributing periodic orbits are affected. The theory is tested numerically using the annulus and the Sinai billiard. For the Sinai billiard the investigation of the spectral density is complemented by an analysis which is based on the scattering approach to quantization. The merits of this approach as a tool to investigate refined semiclassical theories are discussed and demonstrated. 
Harel Primack, Holger Schanz, Uzy Smilansky and Iddo Ussishkin