Uniform distribution of eigenstates on a torus with two point scatterers

Abstract

We study the Laplacian perturbed by two delta potentials on a two-dimensional flat torus. There are two types of eigenfunctions for this operator: old, or unperturbed eigenfunctions which are eigenfunctions of the standard Laplacian, and new, perturbed eigenfunctions which are affected by the scatterers. We prove that along a density one sequence, the new eigenfunctions are uniformly distributed in configuration space, provided that the difference of the scattering points is Diophantine.