Semiclassical inequivalence of polygonalized billiards

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