Superscars for Arithmetic Toral Point Scatterers

Abstract

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

On the phase-space distribution of Bloch eigenmodes for periodic point scatterers

“…These results were partially generalised to three dimensions by Yesha [18,19] who showed that for the cubic torus, all eigenfunctions equidistribute in position space, and that almost all eigenfunctions equidistribute in phase space. These results are further complimented by Kurlberg and Rosenzweig [7] who show the existence of localisation in position representation in 2 dimensions and momentum representation in both 2 and 3 dimensions. In this paper we generalise the results on the cubic torus to include nonzero quasimomentum, which destroys the high eigenvalue multiplicity and consequently, the equidistribution observed by Yesha.…”

“…These results were partially generalised to three dimensions by Yesha [18,19] who showed that for the cubic torus, all eigenfunctions equidistribute in position space, and that almost all eigenfunctions equidistribute in phase space. These results are further complimented by Kurlberg and Rosenzweig [7]…”

Small Scale Equidistribution for a Point Scatterer on the Torus

“…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. The existence of scars for arithmetic point scatterers was established by Kurlberg and Rosenzweig [18]. 1.4.…”

Multifractal eigenfunctions for a singular quantum billiard

“…This system can be rigorously studied by means of von Neumann's theory of self-adjoint extensions. Over the past twenty years much work has been devoted to the study of the spectrum and eigenfunctions of such billiards [12,5,6,7,8,19,31,23,24,21,22]. In particular, arithmetic Seba billiards have been shown to be quantum ergodic [31,23], whereas diophantine Seba billiards possess scarred semiclassical measures [24,21].…”