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

TL;DR

This paper derives explicit formulas for special values of spectral zeta functions associated with certain quantum models, connecting them to classical mathematical constants and integrals, thus advancing understanding of their spectral properties.

Contribution

It provides explicit expressions for spectral zeta function values of quantum Rabi models and non-commutative harmonic oscillators, linking them to classical integrals and irrationality proofs.

Findings

Explicit formulas for spectral zeta values at positive integers.

Connection to Beukers' integral and irrationality of zeta(2).

Extension to two-photon quantum Rabi model.

Abstract

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 Ap\'ery's work. A similar expression of the 1st term of that of 2pQRM is also discussed.