Eigenfunction statistics for a point scatterer on a three-dimensional torus

TL;DR

This paper investigates the distribution of eigenfunctions for a point scatterer on a three-dimensional torus, demonstrating uniform distribution in configuration space for both standard and certain irrational tori.

Contribution

It provides the first analysis of eigenfunction distribution for a point scatterer on a 3D torus, including cases with irrationality conditions.

Findings

Perturbed eigenfunctions are uniformly distributed on the standard torus.

Almost all perturbed eigenfunctions are uniformly distributed on tori with irrationality conditions.

The results extend understanding of quantum chaos and eigenfunction behavior on flat manifolds.

Abstract

In this paper we study eigenfunction statistics for a point scatterer (the Laplacian perturbed by a delta-potential) on a three-dimensional flat torus. The eigenfunctions of this operator are the eigenfunctions of the Laplacian which vanish at the scatterer, together with a set of new eigenfunctions (perturbed eigenfunctions). We first show that for a point scatterer on the standard torus all of the perturbed eigenfunctions are uniformly distributed in configuration space. Then we investigate the same problem for a point scatterer on a flat torus with some irrationality conditions, and show uniform distribution in configuration space for almost all of the perturbed eigenfunctions.