Spectral statistics of rectangular billiards with localized perturbations

Rahav, Saar; Fishman, Shmuel

doi:10.1088/0951-7715/15/5/311

Cited by 25 publications

(40 citation statements)

References 76 publications

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“…An interesting problem would be to understand the strong coupling limit, where λ → ∞ together with φ → π while tan(φ/2) ≈ log λ, so that the RHS of the spectral equation (1.3) blows up. In that range it has been argued that the spectrum displays intermediate statistics [20,5,3,4,17]. Acknowledgments: We thank Maja Rudolph for her help with the numerical investigation of some of these issues, and John Friedlander for discussions concerning sums of two squares.…”

Section: Remarksmentioning

confidence: 99%

Statistics of Wave Functions for a Point Scatterer on the Torus

Rudnick

Ueberschär

2012

Commun. Math. Phys.

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Quantum systems whose classical counterpart have ergodic dynamics are quantum ergodic in the sense that almost all eigenstates are uniformly distributed in phase space. In contrast, when the classical dynamics is integrable, there is concentration of eigenfunctions on invariant structures in phase space. In this paper we study eigenfunction statistics for the Laplacian perturbed by a delta-potential (also known as a point scatterer) on a flat torus, a popular model used to study the transition between integrability and chaos in quantum mechanics. The eigenfunctions of this operator consist of eigenfunctions of the Laplacian which vanish at the scatterer, and new, or perturbed, eigenfunctions. We show that almost all of the perturbed eigenfunctions are uniformly distributed in configuration space.

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Section: Remarksmentioning

confidence: 99%

Statistics of Wave Functions for a Point Scatterer on the Torus

Rudnick

Ueberschär

2012

Commun. Math. Phys.

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“…Billiards with the shapes of rational polygonals containing corners with angles α = π n where n is an integer [14][15][16][17][18][19] and of an integrable one containing pointlike scatterers [20][21][22][23][24][25][26][27][28][29][30] or, in general, an obstacle of a size which is much smaller than the billiard area and smaller or comparable to the wavelength of the quantum particle trapped in it, are pseudo-integrable and almost-integrable, respectively. Pseudo-integrable and almost-integrable systems exhibit an intermediate spectral statistics [31,32] which generally is non-universal.…”

Section: Introductionmentioning

confidence: 99%

Experimental investigation of the fluctuations in nonchaotic scattering in microwave billiards*

Zhang¹,

Zhang²,

Dietz³

et al. 2019

Chinese Phys. B

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We report on the experimental investigation of the properties of the eigenvalues and wavefunctions and the fluctuation properties of the scattering matrix of closed and open billiards, respectively, of which the classical dynamics undergoes a transition from integrable via almost integrable to fully chaotic. To realize such a system we chose a billiard with a 60 • sector shape of which the classical dynamics is integrable, and introduced circular scatterers of varying number, size and position. The spectral properties of generic quantum systems of which the classical counterpart is either integrable or chaotic are universal and well understood. If, however, the classical dynamics is pseudo-integrable or almost-integrable, they exhibit a non-universal intermediate statistics, for which analytical results are known only in a few cases, like, e.g., if it corresponds to semi-Poisson statistics. Since the latter is, above all, clearly distinguishable from those of integrable and chaotic systems our aim was to design a billiard with these features which indeed is achievable by adding just one scatterer of appropriate size and position to the sector billiard. We demonstrate that, while the spectral properties of almostintegrable billiards are sensitive to the classical dynamics, this is not the case for the distribution of the wavefunction components, which was analysed in terms of the strength distribution, and the fluctuation properties of the scattering matrix which coincide with those of typical, fully chaotic systems.

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“…Note that even if the semi-Poisson distribution is an approximation for the NNSD it may not approximate other spectral measures well. For instance, the form factor, which is the Fourier transform of the energy-energy correlation function, satisfies K(0) = 1/2 for the semi-Poisson distribution [7] while for a billiard with point-scatterer one expects to find K(0) = 1 [11]. In particular, for the rectangular billiard with a point-scatterer, the NNSD was computed analytically under some assumptions in [13] and was found not to be given by the semi-Poisson or the Poisson distributions.…”

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confidence: 99%

Point perturbations of circle billiards

Rahav

Richman

Fishman

2003

J. Phys. A: Math. Gen.

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Abstract. The spectral statistics of the circular billiard with a point-scatterer is investigated. In the semiclassical limit, the spectrum is demonstrated to be composed of two uncorrelated level sequences. The first corresponds to states for which the scatterer is located in the classically forbidden region and its energy levels are not affected by the scatterer in the semiclassical limit while the second sequence contains the levels which are affected by the point-scatterer. The nearest neighbor spacing distribution which results from the superposition of these sequences is calculated analytically within some approximation and good agreement with the distribution that was computed numerically is found. Classical dynamics may be illuminating for the understanding of the corresponding quantum mechanical systems. One of the most studied aspects of the relation between classical and quantum mechanics is the connection between the spectral statistics of the quantum system and the dynamical properties of its classical counterpart. Classically integrable systems typically exhibit Poisson-like spectral statistics [1] while classically chaotic systems exhibit spectral statistics of random matrix ensembles [2,3,4,5]. The spectral statistics of integrable and chaotic systems are universal, that is, they do not depend on specific details of the system but rather on the type of motion and its symmetries. There are systems which are intermediate between integrable and chaotic ones and their spectral properties are not known to be universal. Such systems are of experimental relevance. The spectral statistics of mixed systems, for which the phase space is composed of both integrable and chaotic regions, were studied by Berry and Robnik [6]. The spectrum can be viewed as a superposition of uncorrelated level sequences, corresponding to the various regions, which are either chaotic or integrable. The nearest neighbor spacing distribution (NNSD) of such a superposition of sequences was calculated in [6]

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Spectral statistics of rectangular billiards with localized perturbations

Cited by 25 publications

References 76 publications

Statistics of Wave Functions for a Point Scatterer on the Torus

Statistics of Wave Functions for a Point Scatterer on the Torus

Experimental investigation of the fluctuations in nonchaotic scattering in microwave billiards*

Point perturbations of circle billiards

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