Chaos in a one-dimensional integrable quantum system

TL;DR

This paper investigates a one-dimensional quantum system with scale-free point interactions, revealing that despite its integrability, its spectral statistics align with those of chaotic systems described by random matrix theory.

Contribution

It demonstrates that an integrable quantum system can exhibit spectral properties similar to chaotic systems, challenging traditional distinctions between integrable and chaotic quantum spectra.

Findings

Level spacing distribution matches Gaussian Orthogonal Ensemble predictions.

Spectral properties show nontrivial statistical behavior despite integrability.

Explicit secular equation allows detailed spectral analysis.

Abstract

We study a simple one-dimensional quantum system on a circle with n scale free point interactions. The spectrum of this system is discrete and expressible as a solution of an explicit secular equation. However, its statistical properties are nontrivial. The level spacing distribution between its neighboring odd and even levels displays a surprising agreement with the prediction obtained for the Gaussian Orthogonal Ensemble of random matrices.