Special values of spectral zeta functions of one-, two-photon quantum Rabi models and non-commutative harmonic oscillators
Abstract
We find explicit expressions of the special values of the Hurwitz-type spectral zeta function ζ(H;n,λ) for the Hamiltonians 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 Ap\'ery's work. A similar expression of the 1st term of that of 2pQRM is also discussed.
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