Universal spectral bounds for the quantum Rabi model: Extending Braak's conjecture

The quantum Rabi model is a paradigmatic example of a minimal yet nontrivial light-matter interaction, whose spectrum is transcendental yet exhibits a number of regularities. Braak observed that the eigenvalues bunch or anti-bunch following strict rules, leading to a conjecture that links integrability in quantum systems and residual order in their spectra. While a general proof remains elusive, understanding this structure is crucial for distinguishing deterministic quantum dynamics from chaotic behavior. Here, we extend Braak's conjecture through a set of eigenvalue inequalities. We prove the extended conjecture across low and intermediate splitting regimes, and provide universal upper bounds on the entire spectrum. Our results uncover additional layers of spectral organization in the quantum Rabi model and expand the analytic toolkit for strongly coupled quantum systems.