The Quantum Rabi model: Towards Braak's conjecture

TL;DR

This paper proves key spectral properties of the quantum Rabi model, confirming conjectures about its eigenvalue distribution and spectral structure using advanced asymptotic and number theory techniques.

Contribution

It establishes a density one version of Braak's conjecture and confirms a conjecture on spectral spacings for the quantum Rabi model.

Findings

Proves a density one version of Braak's conjecture.

Confirms a conjecture on nearest neighbor spectral spacings.

Uses asymptotic expansion and number theory in the proof.

Abstract

We establish a density one version of Braak's conjecture on the fine structure of the spectrum of the quantum Rabi model, as well as a recent conjecture of Braak, Nguyen, Reyes-Bustos and Wakayama on the nearest neighbor spacings of the spectrum. The proof uses a three-term asymptotic expansion for large eigenvalues due to Boutet de Monvel and Zielinski, and a number theoretic argument from uniform distribution theory.