2006/11/30 by Alex H. Barnett, A. H. Barnett, Timo Betcke +1 · 2 citations
Mathematics · Physics and Astronomy · #Chaotic #Computer science #Dynamical billiards #Eigenvalues and eigenvectors #Geometry #Hierarchy #Mathematics #Phase space #Physics #Quantum #Quantum chaos #Quantum chaos and dynamical systems #Quantum dynamics #Quantum mechanics #Random matrix #Scientific Research and Discoveries #Space (punctuation) #Statistical physics #Theoretical and Computational Physics #nlin.CD
paper · pdf · doi:10.1063/1.2816946
revised, corrected, expanded including new results on dynamical tunneling and level-spacing; 14 pages, 15 figures (4 new), resubmitted to CHAOS
arxiv created 2007/09/07 · openalex publication_date 2007/12/01 · arxiv updated 2009/12/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/05
We report the first large-scale statistical study of very high-lying eigenmodes (quantum states) of the mushroom billiard proposed by L. A. Bunimovich [Chaos 11, 802 (2001)]. The phase space of this mixed system is unusual in that it has a single regular region and a single chaotic region, and no KAM hierarchy. We verify Percival's conjecture to high accuracy (1.7%). We propose a model for dynamical tunneling and show that it predicts well the chaotic components of predominantly regular modes. Our model explains our observed density of such superpositions dying as E(-1/3) (E is the eigenvalue). We compare eigenvalue spacing distributions against Random Matrix Theory expectations, using 16,000 odd modes (an order of magnitude more than any existing study). We outline new variants of mesh-free boundary collocation methods which enable us to achieve high accuracy and high mode numbers (approximately 10(5)) orders of magnitude faster than with competing methods.