Quantum ergodicity for a point scatterer on the three-dimensional torus

TL;DR

This paper proves quantum ergodicity for most eigenfunctions of a point scatterer on a three-dimensional torus, showing they are uniformly distributed in phase space, extending previous results from two dimensions.

Contribution

It establishes quantum ergodicity for perturbed eigenfunctions on a 3D torus, a significant extension of prior 2D results, demonstrating phase space uniform distribution.

Findings

Almost all perturbed eigenfunctions are uniformly distributed in phase space.

The result extends quantum ergodicity to three-dimensional flat tori.

The paper generalizes previous 2D findings to 3D settings.

Abstract

Consider a point scatterer (the Laplacian perturbed by a delta-potential) on the standard three-dimensional flat torus. Together with the eigenfunctions of the Laplacian which vanish at the point, this operator has a set of new, perturbed eigenfunctions. In a recent paper, the author was able to show that all of the perturbed eigenfunctions are uniformly distributed in configuration space. In this paper we prove that almost all of these eigenfunctions are uniformly distributed in phase space, i.e. we prove quantum ergodicity for the subspace of the perturbed eigenfunctions. An analogue result for a point scatterer on the two-dimensional torus was recently proved by Kurlberg and Uebersch\"ar.