Normal modes of billiards portrayed in the stellar (or nodal) representation

“…We define the Husimi transform 48 of functions on R, for convenience reviewing the coherent state formalism in dimensionless ͑ប-free͒ units. Given a width parameter ͑phase space aspect-ratio͒ Ͼ 0, it is easy to show that the annihilation operator …”

Section: Appendix A: Husimi Transformmentioning

confidence: 99%

Quantum mushroom billiards

Barnett

Betcke

2007

Chaos

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We report the first large-scale statistical study of very high-lying eigenmodes ͑quantum states͒ of the mushroom billiard proposed by L. A. Bunimovich ͓Chaos 11, 802 ͑2001͔͒. 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 eigenvalue͒. We compare eigenvalue spacing distributions against Random Matrix Theory expectations, using 16 000 odd modes ͑an order of magnitude more than any existing study͒. We outline new variants of mesh-free boundary collocation methods which enable us to achieve high accuracy and high mode numbers ͑ϳ10 5 ͒ orders of magnitude faster than with competing methods. © 2007 American Institute of Physics. ͓DOI: 10.1063/1.2816946͔ Quantum chaos is the study of the quantum (wave) properties of Hamiltonian systems whose classical (ray) dynamics is chaotic. Billiards are some of the simplest and most studied examples; physically their wave analogs are vibrating membranes, quantum, electromagnetic, or acoustic cavities. They continue to provide a wealth of theoretical challenges. In particular "mixed" systems, where ray phase space has both regular and chaotic regions (the generic case), are difficult to analyze. Six years ago Bunimovich described 1 a mushroom billiard with simple mixed dynamics free of the usual island hierarchies of Kolmogorov-Arnold-Moser (KAM). He concluded by anticipating the growth of "quantum mushrooms;" it is this gardening task that we achieve here, by developing advanced numerical methods to collect an unprecedented large number n of eigenmodes (much higher than competing numerics 2 or microwave studies 3 ). Since uncertainties scale as n −1Õ2 , a large n is vital for accurate spectral statistics and for studying the semiclassical (high eigenvalue) limit. We address three main issues: (i) The conjecture of Percival 4 that semiclassically modes live exclusively in invariant (regular or chaotic) regions, and occur in proportion to the phase space volumes. (ii) The mechanism for dynamical tunneling, or quantum coupling between classically isolated phase space regions. (iii) The distribution of spacings of nearest-neighbor eigenvalues, about which recent questions have been raised. 3 We show many pictures of modes, including the boundary phase space (the so-called Husimi function).

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“…In this paper we will follow the procedure described in Ref. [16], which relies on the use of normal derivatives of the wave functions evaluated at the boundary of the billiard. On this boundary, Birkhoff coordinates [17], (q, p), are used to define both classical and quantal Poincaré surfaces of section, such that q, is the arc length coordinate, and p = p ·t/|p| is the fraction of tangential momentum.…”

Section: Constructing Localized Wave Functionsmentioning

confidence: 99%

Localization properties of groups of eigenstates in chaotic systems

et al. 2001

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In this paper we study in detail the localized wave functions defined in Phys. Rev. Lett. 76, 1613Lett. 76, (1994, in connection with the scarring effect of unstable periodic orbits in highly chaotic Hamiltonian system. These functions appear highly localized not only along periodic orbits but also on the associated manifolds. Moreover, they show in phase space the hyperbolic structure in the vicinity of the orbit, something which translates in configuration space into the structure induced by the corresponding self-focal points. On the other hand, the quantum dynamics of these functions are also studied. Our results indicate that the probability density first evolves along the unstable manifold emanating from the periodic orbit, and localizes temporarily afterwards on only a few, short related periodic orbits. We believe that this type of studies can provide some keys to disentangle the complexity associated to the quantum mechanics of these kind of systems, which permits the construction of a simple explanation in terms of the dynamics of a few classical structures.

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Normal modes of billiards portrayed in the stellar (or nodal) representation

Cited by 58 publications

References 22 publications

Fredholm methods for billiard eigenfunctions in the coherent state representation

Fredholm methods for billiard eigenfunctions in the coherent state representation

Quantum mushroom billiards

Localization properties of groups of eigenstates in chaotic systems

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