Diffractive corrections in the trace formula for polygonal billiards

We derive contributions to the trace formula for the spectral density accounting for the role of diffractive orbits in two-dimensional polygonal billiards. In polygons, diffraction typically occurs at the boundary of a family of trajectories. In this case the first diffractive correction to the contribution of the family to the periodic orbit expansion is of order of that of an isolated orbit, and gives the first sqrt[Planck's over 2pi] correction to the leading semiclassical term. Keller's geometrical theory of diffraction is inadequate for treating these corrections and we develop an alternative approximation based on Kirchhoff's theory. Numerical checks show that our procedure allows reduction of the typical semiclassical error by about two orders of magnitude. The method permits treatment of the related problem of flux-line diffraction with the same degree of accuracy.