Scarred eigenstates for arithmetic toral point scatterers

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.