Scarred eigenstates for arithmetic toral point scatterers

TL;DR

This paper studies eigenfunctions of Laplacians with delta potentials on tori, revealing that despite quantum ergodicity, there is significant scarring in eigenfunctions' phase space representations, especially in 2D.

Contribution

It demonstrates the existence of eigenfunction scarring in phase space for arithmetic toral point scatterers, contrasting with quantum ergodicity results.

Findings

Scarring occurs in momentum representation for both 2D and 3D.

In 2D, eigenfunctions frequently exhibit phase space scarring.

Scarred eigenstates are rare in 3D but common in 2D, with quantitative bounds.

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 there is scarring in the momentum representation for $d=2,3$, as well as in the position representation for $d=2$ (i.e., the eigenfunctions fail to equidistribute in phase space along an infinite subsequence of new eigenvalues.) For $d=3$, scarred eigenstates are quite rare, but for $d=2$ scarring in the momentum representation is very common --- with $N_{2}(x) \sim x/\sqrt{\log x}$ denoting the counting function for the new eigenvalues below $x$, there are $\gg N_{2}(x)/\log^A x$ eigenvalues corresponding to momentum scarred eigenfunctions.