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2006.387: Quantum mushroom billiards

2006.387: Alex H. Barnett and Timo Betcke (2006) Quantum mushroom billiards.

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Abstract

We report the first calculations of eigenmodes (quantum states) of a mushroom billiard of the type proposed by L. Bunimovich in this journal. The phase space of this mixed system has a single regular region and a single ergodic region, and no KAM hierarchy. For a symmetric mushroom with a square foot, we find: i) low-eigenvalue modes with very high relative eigenvalue accuracy of order $10^{-10}$, and ii) high-eigenvalue modes at mode number around $10^5$. We outline the simple but highly-efficient mesh-free boundary collocation methods which make such calculations tractable. We test Percival's conjecture that almost all modes localize either to regular or ergodic regions, report the relative frequencies of such modes, and examine Husimi distributions on the Poincaré surface of section.

Item Type:MIMS Preprint
Additional Information:

submitted to Chaos

Subjects:MSC 2000 > 65 Numerical analysis
MSC 2000 > 81 Quantum theory
MIMS number:2006.387
Deposited By:Dr. Timo Betcke
Deposited On:12 October 2006

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