Weyl's law and quantum ergodicity for maps with divided phase space (with an appendix Converse quantum ergodicity)

Marklof, Jens; O’Keefe, Stephen J.; Zelditch, Steve

doi:10.1088/0951-7715/18/1/015

Cited by 39 publications

(43 citation statements)

References 44 publications

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“…This is supported by several studies, see e.g. [8,9,10,11,12,13]. It is also possible, that the influence of a regular island quantum mechanically extends beyond the outermost invariant curve due to partial barriers like cantori and that quantization conditions remain approximately applicable even outside of the island [8].…”

Section: Introductionmentioning

confidence: 56%

Universality in the flooding of regular islands by chaotic states

2007

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We investigate the structure of eigenstates in systems with a mixed phase space in terms of their projection onto individual regular tori. Depending on dynamical tunneling rates and the Heisenberg time, regular states disappear and chaotic states flood the regular tori. For a quantitative understanding we introduce a random matrix model. The resulting statistical properties of eigenstates as a function of an effective coupling strength are in very good agreement with numerical results for a kicked system. We discuss the implications of these results for the applicability of the semiclassical eigenfunction hypothesis.Comment: 11 pages, 12 figure

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Section: Introductionmentioning

confidence: 56%

Universality in the flooding of regular islands by chaotic states

2007

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“…͑This has been tested in a smooth billiard, 27 and recently proved for certain piecewise linear quantum maps. 28 ͒ We test the conjecture via a matrix element ͑10͒ sensitive to the boundary ͑for numerical efficiency͒; we then can categorize ͑almost all͒ modes as regular or ergodic. We address two issues which have also been raised by recent microwave experiments in the mushroom.…”

Section: Introductionmentioning

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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“…The proof of Quantum Ergodicity [7,45], starting from the ergodicity of the classical map with respect to the Lebesgue measure, is also valid within our nonstandard quantization. Indeed, as shown in [27], the statements i),ii) of Proposition 3.1 and the Egorov theorem (Prop. 3.2) suffice to prove Quantum Ergodicity for the Walsh-quantized baker: Remark 3.…”

Section: Proposition 32 (Egorov Theorem)mentioning

confidence: 72%

Entropy of Semiclassical Measures of the Walsh-Quantized Baker’s Map

Anantharaman

Nonnenmacher

2006

Ann. Henri Poincaré

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We study the baker's map and its Walsh quantization, as a toy model of a quantized chaotic system. We focus on localization properties of eigenstates, in the semiclassical régime. Simple counterexamples show that quantum unique ergodicity fails for this model. We obtain, however, lower bounds on the entropies associated with semiclassical measures, as well as on the Wehrl entropies of eigenstates. The central tool of the proofs is an "entropic uncertainty principle".

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Weyl's law and quantum ergodicity for maps with divided phase space (with an appendix Converse quantum ergodicity)

Cited by 39 publications

References 44 publications

Universality in the flooding of regular islands by chaotic states

Universality in the flooding of regular islands by chaotic states

Quantum mushroom billiards

Entropy of Semiclassical Measures of the Walsh-Quantized Baker’s Map

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