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.

Small Scale Equidistribution for a Point Scatterer on the Torus

“…In [29], it was shown that for a point scatterer on the standard three-dimensional torus T 3 , equidistribution in configuration space holds for all of the new eigenfunctions (and along a density one subsequence of the new eigenfunctions for point scatterers on tori with a Diophantine aspect ratio). Recently, we were able to establish equidistribution in configuration space for tori with two point scatterers [31]. Equidistribution in full phase space (along a density one subsequence) was established both on the standard two-dimensional torus T 2 by Kurlberg and Ueberschär [19], and on the standard three-dimensional torus T 3 [30].…”

Quantum chaos in a harmonic waveguide with scatterers

“…Energy spectra of several physical systems of such type have already been considered. They are the flat rectangular billiardsgeneralization of the Šeba billiard [20] -with 1-3 [22], 6 [23], and 2 [45] scatterers. Theoretical predictions for a single scatterer in a harmonic potential were compared to experiments [56].…”

Quantum chaos in a harmonic waveguide with scatterers

“…Energy spectra of several physical systems of such type have already been considered. They are the flat rectangular billiards -generalization of the Šeba billiard [21] -with 1-3 [23], 6 [24], and 2 [46] scatterers. Theoretical predictions for a single scatterer in a harmonic potential were compared to experiments [57].…”