Statistical analysis of scars in stadium billiard

TL;DR

This paper investigates quantum scars in stadium billiard eigenfunctions across a wide energy range using an improved wave decomposition method, confirming theoretical predictions about scar intensity scaling with Planck's constant.

Contribution

It introduces an enhanced method for analyzing scars in eigenfunctions and provides extensive data supporting the scaling laws of scar intensities.

Findings

Maximal scar intensity scales differently with 7 for various scar types.

Results support existing theories by Bogomolny and Robnik.

Scar profiles are systematically analyzed across many eigenstates.

Abstract

In this paper, by using our improved plane wave decomposition method, we study the scars in the eigenfunctions of the stadium billiard from very low state to as high as about the one millionth state. In the systematic searching for scars of various types, we have used the approximate criterion based on the quantization of the classical action along the unstable periodic orbit supporting the scar. We have analized the profile of the integrated probability density along the orbit. We found that the maximal integrated intensity of different types of scars scales in different way with the $\hbar$, which confirms qualitatively and quantitatively the existing theories of scars such as that of Bogomolny (1988) and that of Robnik (1989).