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 − g2, where N is an integer satisfying a polynomial constraint P_N((2g)2, Δ2) = 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.

References

It is believed that for a given parameter pair $(g,\Delta)$, ($g>0$) there are only finitely many Juddian eigenvalues, though this is not proven.

— The density conjecture for Juddian points for the quantum Rabi model  (2501.12105 - Kumar et al., 21 Jan 2025) in Section 1 (Introduction)

For fixed g,Δ,τ and ρ ∈ {0,1}. Compute the number of non-negative integers N such that the equation P_NN,ρ,τ(x,y) = 0 is satisfied.

— Two-photon quantum Rabi models - Spectral degeneracy and symmetries  (2609.00750 - Reyes-Bustos et al., 1 Sep 2026) in Problem 5.1, Section 5.1, “Questions on the number and distribution of Juddian solutions”

For fixed g,Δ,τ, the number of Juddian solutions of the 2pAQRM is finite. Moreover, it is uniformly bounded with respect to parameters g,Δ,τ.

— Two-photon quantum Rabi models - Spectral degeneracy and symmetries  (2609.00750 - Reyes-Bustos et al., 1 Sep 2026) in Conjecture following Problem 5.1, Section 5.1

For fixed ρ ∈ {0,1}, the family of (g,Δ)-plane curves P_NN,ρ,τ((2g)2,Δ2)=0, is dense in the strip |g|<1/2.

— Two-photon quantum Rabi models - Spectral degeneracy and symmetries  (2609.00750 - Reyes-Bustos et al., 1 Sep 2026) in Conjecture following Figure 5.2, Section 5.1