Localization of eigenfunctions in the stadium billiard

Bies, W. E.; Kaplan, L.; Haggerty, M. R.; Heller, Eric J.

doi:10.1103/physreve.63.066214

Cited by 32 publications

(49 citation statements)

References 25 publications

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“…The resulting statistical distributions for the tunnelling rate agree well with numerically computed ensembles except when the real extension of the tunnelling orbit into the potential well is periodic; in that case, strong deviations from the RMT prediction are observed and it was proposed in [11] that these are due to the effect of scarring on wavefunction statistics as outlined in [13]. Additional evidence in support of this has subsequently been provided in [14].…”

Section: Introductionsupporting

confidence: 68%

“…ζ 0 . The operatorT (z), which is nonunitary, quantises the complex symplectic matrix W (z) defined in (14). Finally, the operator e wâ can be interpreted as a translation along the complex phase-space displacement δζ = −w h 2 1 i .…”

Section: Discussionmentioning

confidence: 99%

“…These distributions have been shown in [11] to describe successfully the tunnelling rate statistics of chaotic double wells when the tunnelling route has a nonperiodic real extension. When the tunnelling route connects to a periodic orbit, however, strong deviations are found [11,14] which we explain in the next sections using the idea of scarring.…”

Section: B Statistics Using Standard Rmtmentioning

confidence: 99%

“…Agreement with the scarred distribution is good and there is marked deviation from the prediction of standard RMT (short-dashed curve). Also shown for comparison is the Porter-Thomas distribution [14] (long-dashed curve).…”

Section: Tunnelling Statistics With Scarringmentioning

confidence: 99%

“…The scarred wavefunction statistics described in the previous section are now used to derive distributions for the anomalous tunnelling rate statistics of [11,14]. The first step is to outline how the discussion for maps in the preceding section may be adapted to the statistics of states in potential-well problems and then we can describe how these ideas are used to construct distributions for tunnelling rates.…”

Section: Tunnelling Statistics With Scarringmentioning

confidence: 99%

See 4 more Smart Citations

Scarring and the Statistics of Tunnelling

Creagh¹,

Lee²,

Whelan

2002

Annals of Physics

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We show that the statistics of tunnelling can be dramatically affected by scarring and derive distributions quantifying this effect. Strong deviations from the prediction of random matrix theory can be explained quantitatively by modifying the Gaussian distribution which describes wavefunction statistics. The modified distribution depends on classical parameters which are determined completely by linearised dynamics around a periodic orbit. This distribution generalises the scarring theory of Kaplan [Phys. Rev. Lett. 80, 2582(1998] to describe the statistics of the components of the wavefunction in a complete basis, rather than overlaps with single Gaussian wavepackets. In particular it is shown that correlations in the components of the wavefunction are present, which can strongly influence tunnellingrate statistics. The resulting distribution for tunnelling rates is tested successfully on a two-dimensional double-well potential.

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Section: Introductionsupporting

confidence: 68%

Section: Discussionmentioning

confidence: 99%

Section: B Statistics Using Standard Rmtmentioning

confidence: 99%

Section: Tunnelling Statistics With Scarringmentioning

confidence: 99%

Section: Tunnelling Statistics With Scarringmentioning

confidence: 99%

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Scarring and the Statistics of Tunnelling

Creagh¹,

Lee²,

Whelan

2002

Annals of Physics

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The Twists and Turns of Modern Science and Technology Established in China in the Twentieth Century

Yangzong¹

2021

Western Influences in the History of Science and Technology in Modern China

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In this paper, we study random features manifested in components of energy eigenfunctions of quantum chaotic systems, given in the basis of unperturbed, integrable systems. Based on semiclassical analysis, particularly on Berry's conjecture, it is shown that the components in classically allowed regions can be regarded as Gaussian random numbers in certain sense, when appropriately rescaled with respect to the average shape of the eigenfunctions. This suggests that, when a perturbed system changes from integrable to chaotic, deviation of the distribution of rescaled components in classically allowed regions from the Gaussian distribution may be employed as a measure for the "distance" to quantum chaos. Numerical simulations performed in the LMG model and the Dicke model show that this deviation coincides with the deviation of the nearest-level-spacing distribution from the prediction of random-matrix theory. Similar numerical results are also obtained in two models without classical counterpart.

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Statistical distribution of components of energy eigenfunctions: from nearly-integrable to chaotic

Wang

2016

Chaos, Solitons & Fractals

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We study the statistical distribution of components in the non-perturbative parts of energy eigenfunctions (EFs), in which main bodies of the EFs lie. Our numerical simulations in five models show that deviation of the distribution from the prediction of random matrix theory (RMT) is useful in characterizing the process from nearly-integrable to chaotic, in a way somewhat similar to the nearest-level-spacing distribution. But, the statistics of EFs reveals some more properties, as described below. (i) In the process of approaching quantum chaos, the distribution of components shows a delay feature compared with the nearest-level-spacing distribution in most of the models studied. (ii) In the quantum chaotic regime, the distribution of components always shows small but notable deviation from the prediction of RMT in models possessing classical counterparts, while, the deviation can be almost negligible in models not possessing classical counterparts. (iii) In models whose Hamiltonian matrices possess a clear band structure, tails of EFs show statistical behaviors obviously different from those in the main bodies, while, the difference is smaller for Hamiltonian matrices without a clear band structure.

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Localization of eigenfunctions in the stadium billiard

Cited by 32 publications

References 25 publications

Scarring and the Statistics of Tunnelling

Scarring and the Statistics of Tunnelling

The Twists and Turns of Modern Science and Technology Established in China in the Twentieth Century

Statistical distribution of components of energy eigenfunctions: from nearly-integrable to chaotic

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