Semiclassical inequivalence of polygonalized billiards

Polygonalization of any smooth billiard boundary can be carried out in seve ral ways. We show here that the semiclassical description depends on the po lygonalization process and the results can be inequivalent. We also establi sh that generalized tangent polygons are closest to the corresponding smoot h billiard and for de Broglie wavelengths larger than the average length of the edges, the two are semiclassically equivalent.