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We study diffraction corrections to the semiclassical spectral density ofdispersing (Sinai) billiards. They modify the contributions of periodic orbits(PO's), with at least one segment which is almost tangent to the concave partof the boundary. Given a wavenumber $k$, all the PO's with length up to theHeisenberg length $O(k)$ are required for quantization. We show that most ofthe contributions of PO's which are longer than a limit $O(k^{2/3})$ must becorrected for diffraction effects. For orbits which just miss tangency, thecorrections are of the same magnitude as the semiclassical contributionsthemselves. Orbits which bounce at extreme forward angles give very small termsin the standard semiclassical theory. The diffraction corrections increasetheir amplitude substantially.Comment: 12 pages, self-extracting uuencoded-tar-compressed PostScript file.  In case of extraction problems, please write to Harel Primack at  fnharel1@wicc.weizmann.ac.i

Penumbra Diffraction in the Quantization of Dispersing Billiards