Quantum mushroom billiards

Barnett, Alex H. and Betcke, Timo (2007) Quantum mushroom billiards. Chaos, 17 (4). 043125. ISSN 1749-9097

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Abstract

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 (~105) orders of magnitude faster than with competing methods.

Item Type:	Article
Subjects:	MSC 2010, the AMS's Mathematics Subject Classification > 65 Numerical analysis MSC 2010, the AMS's Mathematics Subject Classification > 81 Quantum theory
Depositing User:	Dr. Timo Betcke
Date Deposited:	07 Apr 2008
Last Modified:	20 Oct 2017 14:12
URI:	https://eprints.maths.manchester.ac.uk/id/eprint/1074

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• Quantum mushroom billiards. (deposited 12 Oct 2006)
  • Quantum mushroom billiards. (deposited 03 Oct 2007)
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