Coupling of bouncing-ball modes to the chaotic sea and their counting function

TL;DR

This paper investigates how bouncing-ball modes in two-dimensional billiards couple to chaotic modes, predicting decay rates analytically and confirming the asymptotic behavior of their counting function through numerical analysis.

Contribution

It introduces an analytical approach to predict decay rates of bouncing-ball modes and determines their asymptotic counting function in specific billiard systems.

Findings

Agreement between analytical decay rates and numerical results for stadium and cosine billiards.

Derived asymptotic exponent =5/8 for cosine billiard, below previous upper bound.

Confirmed the =3/4 exponent for stadium billiard.

Abstract

We study the coupling of bouncing-ball modes to chaotic modes in two-dimensional billiards with two parallel boundary segments. Analytically, we predict the corresponding decay rates using the fictitious integrable system approach. Agreement with numerically determined rates is found for the stadium and the cosine billiard. We use this result to predict the asymptotic behavior of the counting function N_bb(E) ~ E^\delta. For the stadium billiard we find agreement with the previous result \delta = 3/4. For the cosine billiard we derive \delta = 5/8, which is confirmed numerically and is well below the previously predicted upper bound \delta=9/10.