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

TL;DR

This paper presents a method to construct high energy semiclassical wave functions in billiards, like the Bunimovich stadium, using periodic orbits and polygon approximations, naturally incorporating phenomena like scars and superscars.

Contribution

It introduces a novel approach to approximate semiclassical wave functions in billiards via polygonal models based on periodic orbits, including scars and superscars.

Findings

Wave functions approximated by polygonal billiards derived from periodic orbits.

Inclusion of scars and superscars as limits of periodic orbit channels.

Application demonstrated on Bunimovich stadium billiards.

Abstract

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.