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

Abstract

We estimate the error in the semiclassical trace formula for the Sinai billiard under the assumption that the largest source of error is due to Penumbra diffraction, that is diffraction effects for trajectories passing within a distance R O((kR)^(-2/3) to the disk and trajectories being scattered in very forward directions. Here k is the momentum and R the radius of the scatterer. The semiclassical error is estimated by perturbing the Berry-Keating formula. The analysis necessitates an asymptotic analysis of very long periodic orbits. This is obtained within an approximation originally due to Baladi Eckmann and Ruelle. We find that the average error, for sufficiently large value of kR, will exceed the mean level spacing.