Statistics of wave functions for a point scatterer on the torus

TL;DR

This paper investigates how eigenfunctions of a Laplacian with a point scatterer on a flat torus distribute in space, revealing that most are uniformly spread despite the classical system's integrability.

Contribution

It demonstrates that, for a quantum system with a point scatterer on a torus, most perturbed eigenfunctions are uniformly distributed, bridging understanding between integrable and chaotic quantum systems.

Findings

Most perturbed eigenfunctions are uniformly distributed in configuration space.

Eigenfunctions vanish at the scatterer, while perturbed ones do not.

The study models the transition between integrability and chaos in quantum mechanics.

Abstract

Quantum systems whose classical counterpart have ergodic dynamics are quantum ergodic in the sense that almost all eigenstates are uniformly distributed in phase space. In contrast, when the classical dynamics is integrable, there is concentration of eigenfunctions on invariant structures in phase space. In this paper we study eigenfunction statistics for the Laplacian perturbed by a delta-potential (also known as a point scatterer) on a flat torus, a popular model used to study the transition between integrability and chaos in quantum mechanics. The eigenfunctions of this operator consist of eigenfunctions of the Laplacian which vanish at the scatterer, and new, or perturbed, eigenfunctions. We show that almost all of the perturbed eigenfunctions are uniformly distributed in configuration space.