Uniform Distribution of Eigenstates on a Torus with Two Point Scatterers

TL;DR

This paper proves that on a 2D torus with two point scatterers, the new eigenfunctions become uniformly distributed in space along a dense sequence, assuming the scatterers' positions satisfy a Diophantine condition.

Contribution

It establishes the uniform distribution of perturbed eigenfunctions on a torus with two scatterers, under Diophantine conditions, advancing understanding of quantum chaos and eigenfunction behavior.

Findings

New eigenfunctions are uniformly distributed along a density one sequence.

Uniform distribution holds when the difference of scatterer positions is Diophantine.

Results contribute to quantum chaos and eigenfunction distribution theory.

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.