Semiclassical approximations to diffractive effects in the annulus billiard

The semiclassical theory formulated by Bogomolny [E. B. Bogomolny, Nonlinearity 5, 805 (1992)] employs a transfer operator constructed from classical trajectories that connect points on a suitably chosen Poincarésurface of section. In this paper we study the two-dimensional annulus billiard, and modify Bogomolny's transfer operator to include diffractive paths. The penumbra contributions [H. Primack et al., Phys. Rev. Lett. 76, 1615 (1996)], which correspond to diffractive paths passing close to the inner circle of the annulus, are found to account for most of the difference between the exact transfer operator and Bogomolny's semiclassical transfer operator. When these diffractive effects are included, the semiclassical energy eigenvalues are brought much closer to the exact quantum energy eigenvalues.