Finiteness of Juddian eigenvalues for fixed parameters in the Quantum Rabi model

Determine whether, for any fixed coupling g > 0 and level splitting Δ > 0, the Quantum Rabi Hamiltonian H_{g,Δ} admits only finitely many Juddian (degenerate exceptional) eigenvalues; equivalently, ascertain whether there exist only finitely many integers N for which the constraint polynomial P_N((2g)^2, Δ^2) = 0 holds, yielding Juddian eigenvalues E = N − g^2.

Background

The Quantum Rabi Hamiltonian H_{g,Δ} models a two-level atom interacting with a single-mode quantized field, with coupling g and level splitting 2Δ. Doubly degenerate exceptional (Juddian) eigenvalues occur only at special parameter values and correspond to eigenvalues E = N − g^2, where N is an integer satisfying a polynomial constraint P_N((2g)^2, Δ^2) = 0.

While the paper proves a strong form of the density conjecture (showing the existence of a dense set of couplings g producing Juddian eigenvalues for fixed Δ) and constructs infinitely many parameter pairs with two distinct Juddian eigenvalues, the question of whether for a given fixed pair (g, Δ) there are only finitely many Juddian eigenvalues remains unresolved. Establishing finiteness would clarify the global structure of Juddian points for fixed parameters and complement the density result proven in the paper.