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Communications in Mathematical Physics

, Volume 349, Issue 1, pp 329–360 | Cite as

Superscars for Arithmetic Toral Point Scatterers

  • Pär Kurlberg
  • Lior RosenzweigEmail author
Open Access
Article

Abstract

We investigate eigenfunctions of the Laplacian perturbed by a delta potential on the standard tori \({\mathbb{R}^d/2\pi\mathbb{Z}^d}\) in dimensions \({d = 2,3}\). Despite quantum ergodicity holding for the set of “new” eigenfunctions we show that superscars occur—there is phase space localization along families of closed orbits, in the sense that some semiclassical measures contain a finite number of Lagrangian components of the form \({c_{i} \cdot dx\delta({\xi}-{\xi}_{i})}\), for \({c_{i} > 0}\) uniformly bounded from below. In particular, for both \({d = 2}\) and \({d = 3}\), eigenfunctions fail to equidistribute in phase space along an infinite subsequence of new eigenvalues. For \({d = 2}\), we also show that some semiclassical measures have both strongly localized momentum marginals and non-uniform quantum limits (i.e., the position marginals are non-uniform). For \({d = 3}\), superscarred eigenstates are quite rare, but for \({d = 2}\) we show that the phenomenon is quite common—with \({N_{2}(x) \sim x/\sqrt{\log x}}\) denoting the counting function for the new eigenvalues below x, there are \({\gg N_{2}(x)/{\rm log}^A x}\) eigenvalues \({\lambda}\) with the property that any semiclassical limit along these eigenvalues exhibits superscarring.

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© The Author(s) 2016

Open AccessThis article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.

Authors and Affiliations

  1. 1.Department of MathematicsKTH Royal Institute of TechnologyStockholmSweden

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