Penumbra Diffraction in the Quantization of Dispersing Billiards

Abstract

Diffraction corrections to the semiclassical spectral density of dispersing (Sinai) billiards, due to orbits which are almost tangent to the concave part of the boundary, are studied here for the first time. We show that most periodic orbits needed for quantization must be corrected. For orbits which just miss tangency, the corrections are of the same magnitude as the semiclassical contributions themselves. For orbits which glance at an extreme forward direction, the new theory replaces the semiclassical term that approaches $0$ at tangency with a finite one. These corrections are one of the most significant modifications of the trace formula considered so far.