The density conjecture for Juddian points for the quantum Rabi model

TL;DR

This paper proves that for the quantum Rabi model, Juddian eigenvalues are densely present across various coupling strengths, revealing intricate spectral properties linked to Laguerre polynomial zeros.

Contribution

It establishes a strong form of the density conjecture for Juddian points in the quantum Rabi model, connecting eigenvalue degeneracies to polynomial zero structures.

Findings

Dense set of coupling strengths with Juddian eigenvalues for fixed atomic level splitting

Construction of parameter sets with multiple Juddian eigenvalues

Role of Laguerre polynomial zeros in spectral analysis

Abstract

We study doubly degenerate (Juddian) eigenvalues for the Quantum Rabi Hamiltonian, a simple model of the interaction between a two-level atom and a single quantized mode of light. We prove a strong form of the density conjecture of Kimoto, Reyes-Bustos, and Wakayama, showing that any fixed value of the splitting between the two atomic levels, there is a dense set of coupling strengths for which the corresponding Rabi Hamiltonian admits Juddian eigenvalues. We also construct infinitely many sets of parameters for which the Rabi Hamiltonian admits two distinct Juddian eigenvalues. The fine structure of the zeros of classical Laguerre polynomials plays a key role in our methods.