Percival's Conjecture for the Bunimovich Mushroom Billiard

TL;DR

This paper proves Percival's conjecture for a family of mushroom billiards, showing that eigenfunctions split into integrable and ergodic regions, providing the first known example satisfying the conjecture.

Contribution

It demonstrates that mushroom billiards satisfy Percival's conjecture for almost all parameters, offering the first explicit example of such a billiard.

Findings

Proves Percival's conjecture for mushroom billiards.

Shows eigenfunctions split into integrable and ergodic regions.

Provides the first explicit example satisfying Percival's conjecture.

Abstract

The Laplace-Beltrami eigenfunctions on a compact Riemannian manifold $M$ whose geodesic billiard flow has mixed character have been conjectured by Percival to split into two complementary families, with all semiclassical mass supported in the completely integrable and ergodic regions of phase space respectively. In this paper, we consider the Dirichlet Laplacian on a family of mushroom billiards $M_t$ parametrised by the length $t\in(0,2]$ of their rectangular part. We prove that $M_t$ satisfies Percival's conjecture for almost all $t\in(0,2]$, hence providing the first example of a billiard known to satisfy Percival's conjecture.