Special values of spectral zeta functions of one-, two-photon quantum Rabi models and non-commutative harmonic oscillators

We find explicit expressions of the special values of the Hurwitz-type spectral zeta function $ζ(\mathrm{H};n,λ)$ for the Hamiltonians $\mathrm{H}$ of the one-photon quantum Rabi model (1pQRM), the two-photon quantum Rabi model (2pQRM), and the non-commutative harmonic oscillator (NCHO), at positive integers $n$. Then the 1st term of the spectral zeta function of 1pQRM gives a generalization of Beukers’ integral used for the proof of the irrationality of $ζ(2)$ after Apéry’s work. A similar expression of the 1st term of that of 2pQRM is also discussed.

💡 Research Summary

The paper investigates Hurwitz‑type spectral zeta functions ζ(H; s, λ) associated with three quantum‑mechanical operators: the one‑photon quantum Rabi model (1pQRM), the two‑photon quantum Rabi model (2pQRM), and the non‑commutative harmonic oscillator (NCHO). After a careful renormalization of the Hamiltonians—adding constant shifts for the 1pQRM, applying a hyperbolic‑function rescaling for the 2pQRM, and a scaling factor for the NCHO—the authors obtain operators eH(g,Δ,ε) that possess purely discrete spectra under suitable parameter constraints (|Δ| small enough, αβ>1, etc.). For a discrete spectrum {μ_j}, the Hurwitz‑type spectral zeta function is defined as ζ(H; s, λ)=∑_j(μ_j+λ)^{−s}=Tr