High energy semiclassical wave functions in the Bunimovich stadium billiards determined by its periodic orbits

It is argued that the high energy semiclassical wave functions (SWF) in an arbitrary billiards can be built by approximating the billiards by a respective polygon one. The latter billiards is determined by a finite number of periodic orbits of the original one limited by their lengths beginning with the shortest ones and which are common for both the billiards. The phenomenon of scars and superscars (Heller, E.J., {\it Phys. Rev. Lett.} {\bf 53},(1984) 1515) are then naturally incorporated into such a construction being a limit of periodic orbit channels (POCs) considered by Bogomolny and Schmit ({\it Phys. Rev. Lett.} {\bf 92} (2004) 244102). The Bunimovich stadium billiards is considered as an example of such an approach.