Superscars for Arithmetic Toral Point Scatterers

Pär Kurlberg,Lior Rosenzweig

doi:10.1007/s00220-016-2749-x

Pär Kurlberg, Lior Rosenzweig

Open Access

https://doi.org/10.1007/s00220-016-2749-x

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Abstract

We investigate eigenfunctions of the Laplacian perturbed by a delta potential on the standard tori {mathbb{R}^d/2pimathbb{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 dxdelta({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.

Highlights

A basic question in Quantum Chaos is the classification of quantum limits of energy eigenstates of quantized Hamiltonians
With {ψλ}λ denoting Laplace eigenfunctions giving an orthonormal basis for L2(M), a quantum limit is a weak∗ limit of |ψλ(x)|2 along any subsequence of eigenvalues λ tending to infinity
[37] Šeba proposed quantum billiards on rectangles with irrational aspect ratio, perturbed with a delta potential, as a solvable singular model exhibiting wave chaos; in particular that the level spacings should be given by random matrix theory (GOE)

Summary

Introduction

A basic question in Quantum Chaos is the classification of quantum limits of energy eigenstates of quantized Hamiltonians. We briefly review some results and definitions about point scatterers and give a short number theoretic background. We begin with the point scatterers, and recall the definition and properties of the quantization of observables (see [27,35] for more details; further background can be found in [43,45]). The restriction is symmetric though not self-adjoint, but by von Neumann’s theory of self adjoint extensions there exists a one-parameter family of self-adjoint extensions; for φ ∈ Eigenvalues of the unperturbed Laplacian, and the corresponding eigenfunctions that vanish at x0. The multiplicities of the new eigenvalues are reduced by 1, due to the constraint of vanishing at x0. New eigenvalues λ ∈ R satisfying the equation rd (n)

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