Statistics of Wave Functions for a Point Scatterer on the Torus

Rudnick, Zeév; Ueberschär, Henrik

doi:10.1007/s00220-012-1556-2

Cited by 29 publications

(60 citation statements)

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“…The first principle result asserts the small scale equidistribution for the new eigenfunctions of a point scatterer on the standard flat twodimensional torus T 2 , holding (almost) all the way down to the Planck scale. In particular, we significantly strengthen the main result in [22] in this case. Theorem 1.1.…”

Section: Statement Of the Main Resultssupporting

confidence: 82%

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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Section: Statement Of the Main Resultssupporting

confidence: 82%

Small Scale Equidistribution for a Point Scatterer on the Torus

Yesha

2020

Commun. Math. Phys.

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“…We have the following lemma which can be found in the standard literature on point scatterers, or in the appendix to the article by Rudnick & Ueberschär [7]. …”

Section: (B) the Quantization Conditionmentioning

confidence: 99%

“…We have the following theorem, which is proved in [7]. In the paper, the theorem is stated for the specific case of the eigenfunctions g λ ϕ j of the weakly coupled point scatterer.…”

Section: Statistics Of Wave Functionsmentioning

confidence: 99%

Quantum chaos for point scatterers on flat tori

Ueberschär

2014

Phil. Trans. R. Soc. A.

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“…(see equation (5.2) of [37]). Since the torus is homogeneous we may without loss of generality assume that x 0 = 0.…”

Section: Introductionmentioning

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%

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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Statistics of Wave Functions for a Point Scatterer on the Torus

Cited by 29 publications

References 20 publications

Small Scale Equidistribution for a Point Scatterer on the Torus

Small Scale Equidistribution for a Point Scatterer on the Torus

Quantum chaos for point scatterers on flat tori

Superscars for Arithmetic Toral Point Scatterers

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