Error of semiclassical eigenvalues in the semiclassical limit - an asymptotic analysis of the Sinai billiard

We estimate the error in the semiclassical trace formula for the Sinai bill iard under the assumption that the largest source of error is due to penumb ra diffraction: namely, diffraction effects for trajectories passing within a distance R.O((kR)(-2/3)) to the disc and trajectories being scattered in very forward directions. Here k is the momentum and R the radius of the sc atterer. The semiclassical error is estimated by perturbing the Berry-Keati ng formula. The analysis necessitates an asymptotic analysis of very long p eriodic orbits. This is obtained within an approximation originally due to Baladi, Eckmann and Ruelle. We find that the average error, for sufficientl y large values of kR, will exceed the mean level spacing.