Localized eigenfunctions in Šeba billiards

Keating, Jon P; Marklof, Jens; Winn, Brian

doi:10.1063/1.3393884

Cited by 10 publications

(15 citation statements)

References 39 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…Keating et al [19] prove under assumptions on the spectrum (which are consistent with the Berry-Tabor conjecture) that there exists a positive density subsequence of eigenfunctions for the Šeba billiard (weak coupling) which become localized around two Laplacian eigenfunctions. In particular, it follows that this subsequence becomes localized in the high energy limit, therefore disproving quantum ergodicity.…”

Section: Remark 42mentioning

confidence: 77%

Quantum chaos for point scatterers on flat tori

Ueberschär

2014

Phil. Trans. R. Soc. A.

View full text Add to dashboard Cite

show abstract

Section: Remark 42mentioning

confidence: 77%

Quantum chaos for point scatterers on flat tori

Ueberschär

2014

Phil. Trans. R. Soc. A.

View full text Add to dashboard Cite

show abstract

“…A related, and in some sense complementary, issue was studied by Berkolaiko, Keating and Winn [2] who predict that for an irrational torus with a point scatterer there is a subsequence of eigenfuctions which "scar" in momentum space, and this was proved by Keating, Marklof and Winn [11] to be the case assuming that the eigenvalues of the Laplacian on the unperturbed irrational torus have Poisson spacing distribution, as is predicted by the Berry-Tabor conjecture.…”

Section: Remarksmentioning

confidence: 89%

Statistics of Wave Functions for a Point Scatterer on the Torus

Rudnick

Ueberschär

2012

Commun. Math. Phys.

View full text Add to dashboard Cite

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.

show abstract

“…Further, these type of scars persist on adding certain perturbations that destroy the spectral multiplicities [24]. Other models where scarring is known to exist include toral point scatterers with irrational aspect ratios [29,22,3] and quantum star graphs [4], though neither model is quantum ergodic [29,4].…”

Section: Following Kurlberg and Wigmanmentioning

confidence: 99%

Superscars for Arithmetic Toral Point Scatterers

Kurlberg

Rosenzweig

2016

Commun. Math. Phys.

View full text Add to dashboard Cite

We consider momentum push-forwards of measures arising as quantum limits (semiclassical measures) of eigenfunctions of a point scatterer on the standard flat torus T 2 = R 2 /Z 2 . Given any probability measure arising by placing delta masses, with equal weights, on Z 2 -lattice points on circles and projecting to the unit circle, we show that the mass of certain subsequences of eigenfunctions, in momentum space, completely localizes on that measure and are completely delocalized in position (i.e., concentration on Lagrangian states.) We also show that the mass, in momentum, can fully localize on more exotic measures, e.g. singular continous ones with support on Cantor sets. Further, we can give examples of quantum limits that are certain convex combinations of such measures, in particular showing that the set of quantum limits is richer than the ones arising only from weak limits of lattice points on circles. The proofs exploit features of the half-dimensional sieve and behavior of multiplicative functions in short intervals.Date: December 16, 2019. 1 probability measures on the unit circle, is of the form cµ sing + (1 − c)µ uniform , for some c ∈ [1/2, 1]. Here both µ uniform and µ sing are normalized to have mass one, and µ sing can be taken to be a sum of delta measures giving equal mass to the four points ±(1, 0), ±(0, 1). We note that µ uniform is the push-forward of the Liouville measure and hence maximally delocalized, whereas µ sing is maximally localized since any quantum limits in this setting must be invariant under a certain eight fold symmetry (cf. (1.5)).Stronger localization, i.e., going strictly beyond c = 1/2, is particularly interesting given a number of "half delocalization" results for quantum limits for some other (strongly chaotic) systems, namely quantized cat maps and geodesic flows on manifolds with constant negative curvature. For example, in the former case Faure and Nonnenmacher showed [12] that if a quantum limit ν is decomposed as ν = ν pp + ν Liouville + ν sc , with ν pp denoting the pure point part and ν sc denoting the singular continous part, then ν Liouville (T 2 ) ≥ ν pp (T 2 ), and thus ν pp (T 2 ) ≤ 1/2. (We emphasize that T 2 is the full phase space in this setting.)The aim of this paper is to exhibit strong, essentially maximal, localization for a quantum ergodic system, namely arithmetic toral point scatterers. In particular we construct quantum limits (in momentum) corresponding to c = 1 in the above decomposition; other interesting examples include singular continous measures with support, say, on Cantor sets. This can be viewed as a step towards a "measure classification" for quantum limits of quantum ergodic systems.1.1. Description of the model. Let us now describe the basic properties of the point scatterer. This is discussed in further detail in [37,38,28,26,39,41]. To describe the quantum system associated with the point scatterer, consider −∆| Dx 0 where 5

show abstract

Localized eigenfunctions in Šeba billiards

Cited by 10 publications

References 39 publications

Quantum chaos for point scatterers on flat tori

Quantum chaos for point scatterers on flat tori

Statistics of Wave Functions for a Point Scatterer on the Torus

Superscars for Arithmetic Toral Point Scatterers

Contact Info

Product

Resources

About