Quantum Ergodicity for Point Scatterers on Arithmetic Tori

Kurlberg, Pär; Ueberschär, Henrik

doi:10.1007/s00039-014-0275-6

Cited by 14 publications

(33 citation statements)

References 19 publications

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“…We note that the quantization of our observables is as explicitly given in (5.1), which follows the approach of [28].…”

Section: Following Kurlberg and Wigmanmentioning

confidence: 99%

“…Combining the two estimates above completes the proof. Following Kurlberg and Ueberschär [28], we quantize our observables as follows. For g ∈ L 2 (S 1 ) let and for ξ = 0, f ( ξ |ξ| ) is defined to be S 1 f (θ) dθ 2π .…”

Section: For Nmentioning

confidence: 99%

“…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…”

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confidence: 99%

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Superscars for Arithmetic Toral Point Scatterers

Kurlberg

Rosenzweig

2016

Commun. Math. Phys.

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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

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“…We note that the quantization of our observables is as explicitly given in (5.1), which follows the approach of [28].…”

Section: Following Kurlberg and Wigmanmentioning

confidence: 99%

Section: For Nmentioning

confidence: 99%

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Superscars for Arithmetic Toral Point Scatterers

Kurlberg

Rosenzweig

2016

Commun. Math. Phys.

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“…Recently, we were able to establish equidistribution in configuration space for tori with two point scatterers [31]. Equidistribution in full phase space (along a density one subsequence) was established both on the standard two-dimensional torus T 2 by Kurlberg and Ueberschär [19], and on the standard three-dimensional torus T 3 [30]. The quantum limits of a point scatterer on a torus with an irrational aspect ratio (also known as the Šeba billiard [23]) were further studied by Kurlberg-Ueberschär [20], who proved the existence of "scars", i.e., localized quantum limits.…”

Section: Toral Point Scatterersmentioning

confidence: 99%

Small Scale Equidistribution for a Point Scatterer on the Torus

Yesha

2020

Commun. Math. Phys.

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We study the small scale distribution of the eigenfunctions of a point scatterer (the Laplacian perturbed by a delta potential) on two-and three-dimensional flat tori. In two dimensions, we establish small scale equidistribution for the "new" eigenfunctions holding all the way down to the Planck scale. In three dimensions, small scale equidistribution is established for all of the "new" eigenfunctions at certain scales.

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“…In dimension 2, if we denote by σ(∆) the integers which are a sum of two squares (equivalently the set of Laplace eigenvalues), then, according to [KuUe14,Cor. 3.6], one can find, for every δ > 0, a density 1 subset S δ of σ(∆) such that, for every λ 2 ∈ S δ ,…”

Section: Proof Of the Variance Estimatesmentioning

confidence: 99%

Equidistribution of toral eigenfunctions along hypersurfaces

Hezari

Rivière

2019

Rev. Mat. Iberoam.

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We prove a new quantum variance estimate for toral eigenfunctions. As an application, we show that, given any orthonormal basis of toral eigenfunctions and any smooth embedded hypersurface with nonvanishing principal curvatures, there exists a density one subsequence of eigenfunctions that equidistribute along the hypersurface. This is an analogue of the Quantum Ergodic Restriction theorems in the case of the flat torus, which in particular verifies the Bourgain-Rudnick's conjecture on L 2 -restriction estimates for a density one subsequence of eigenfunctions in any dimension. Using our quantum variance estimates, we also obtain equidistribution of eigenfunctions against measures whose supports have Fourier dimension larger than d − 2. In the end, we also describe a few quantitative results specific to dimension 2.

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Quantum Ergodicity for Point Scatterers on Arithmetic Tori

Cited by 14 publications

References 19 publications

Superscars for Arithmetic Toral Point Scatterers

Superscars for Arithmetic Toral Point Scatterers

Small Scale Equidistribution for a Point Scatterer on the Torus

Equidistribution of toral eigenfunctions along hypersurfaces

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