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 $\zeta (\mathrm{H};n,\lambda )$ 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 $\zeta (2)$ after Apéry’s work. A similar expression of the 1st term of that of 2pQRM is also discussed.