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.

31 pages, 1 figure

References in corpus (4)

• Eigenfunction statistics for a point scatterer on a three-dimensional torus
• Quantum Chaos for point scatterers on flat tori
• On the eigenvalue spacing distribution for a point scatterer on the flat torus
• On the Phase-Space Distribution of Bloch Eigenmodes for Periodic Point Scatterers

Cited by in corpus (3)

• Multifractal eigenfunctions for a singular quantum billiard
• Small scale equidistribution for a point scatterer on the torus
• On the Phase-Space Distribution of Bloch Eigenmodes for Periodic Point Scatterers