Chaotic billiards generated by arithmetic groups

Bogomolny, E.; Georgeot, Bertrand; Giannoni, M.J.; Schmit, C.

doi:10.1103/physrevlett.69.1477

Cited by 97 publications

(94 citation statements)

References 12 publications

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“…This is hardly surprising; Sarnak [44] notes that a simple reflection symmetry is enough to cause g = 0. Arithmetic systems are very special and have many symmetries: all periodic orbits possess a single Lyapunov exponent, and eigenvalue spacing statistics are unusual for ergodic systems [16]. This makes the study of a planar Euclidean billiard system without symmetry (where Lyapunov exponents differ), for which no number-theoretic analytic tools exist, particularly interesting.…”

Section: Introductionmentioning

confidence: 99%

Asymptotic rate of quantum ergodicity in chaotic Euclidean billiards

Barnett

2006

Comm. Pure Appl. Math.

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The Quantum Unique Ergodicity (QUE) conjecture of RudnickSarnak is that every eigenfunction φ n of the Laplacian on a manifold with uniformly-hyperbolic geodesic flow becomes equidistributed in the semiclassical limit (eigenvalue E n → ∞), that is, 'strong scars' are absent. We study numerically the rate of equidistribution for a uniformly-hyperbolic Sinai-type planar Euclidean billiard with Dirichlet boundary condition (the 'drum problem') at unprecedented high E and statistical accuracy, via the matrix elements φ n ,Âφ m of a piecewise-constant test function A. By collecting 30000 diagonal elements (up to level n ≈ 7 × 10 5 ) we find that their variance decays with eigenvalue as a power 0.48±0.01, close to the estimate 1/2 of FeingoldPeres (FP). This contrasts the results of existing studies, which have been limited to E n a factor 10 2 smaller. We find strong evidence for QUE in this system. We also compare off-diagonal variance, as a function of distance from the diagonal, against FP at the highest accuracy (0.7%) thus far in any chaotic system. We outline the efficient scaling method used to calculate eigenfunctions.

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Section: Introductionmentioning

confidence: 99%

Asymptotic rate of quantum ergodicity in chaotic Euclidean billiards

Barnett

2006

Comm. Pure Appl. Math.

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“…While some obvious counter-examples exist, such as the sphere in the integrable case (the levels are k(k + 1) with multiplicity 2k + 1), and more subtle examples in the chaotic case, such as the modular surface (the quotient of the upper half-plane by the modular group SL(2, Z)), where the spacings appear to be Poissonian [1], [6], [13], [7], there is sufficient numerical evidence for us to believe that these universality conjectures hold in the generic case.…”

Section: Introductionmentioning

confidence: 99%

Eigenvalue Spacings for Regular Graphs

Jakobson

Miller

Rivin

et al. 1999

Emerging Applications of Number Theory

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“…However, exceptions are well documented in the literature, among others we cite arithmetic billiards [5], Harper model [6], and the ubiquitous kicked rotor [7], namely, a free particle in a circle which experiences periodic kicks of amplitude given by a periodic smooth potential,…”

Section: Introductionmentioning

confidence: 99%

Anderson Localization in Quantum Chaos: Scaling and Universality

García-García

Wang

2007

Acta Phys. Pol. A

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The one-parameter scaling theory is a powerful tool to investigate Anderson localization effects in disordered systems. In this paper we show that this theory can be adapted to the context of quantum chaos provided that the classical phase space is homogeneous, not mixed. The localization problem in this case is defined in momentum, not in real space. We then employ the one-parameter scaling theory to: (a) propose a precise characterization of the type of classical dynamics related to the Wigner-Dyson and Poisson statistics which also predicts in which situations Anderson localization corrections invalidate the relation between classical chaos and random matrix theory encoded in the Bohigas-Giannoni-Schmit conjecture, (b) to identify the universality class associated with the metal-insulator transition in quantum chaos. In low dimensions it is characterized by classical superdiffusion, in higher dimensions it has in general a quantum origin as in the case of disordered systems. We illustrate these two cases by studying 1d kicked rotors with non-analytical potentials and a 3d kicked rotor with a smooth potential.

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Chaotic billiards generated by arithmetic groups

Cited by 97 publications

References 12 publications

Asymptotic rate of quantum ergodicity in chaotic Euclidean billiards

Asymptotic rate of quantum ergodicity in chaotic Euclidean billiards

Eigenvalue Spacings for Regular Graphs

Anderson Localization in Quantum Chaos: Scaling and Universality

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