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

TL;DR

The paper derives explicit expressions for the special values $\zeta(H;n,\lambda)$ of Hurwitz-type spectral zeta functions for renormalized one-photon QRM, two-photon QRM, and non-commutative harmonic oscillator Hamiltonians, expressing them as a sum of Hurwitz terms and an infinite series in the perturbation parameter that involves derivatives of trace-like functions $R_m$. It develops integral representations for these $R_m$ terms via Lie semigroup methods and then specializes to the first nontrivial term ($m=1$), revealing Beukers-type integrals and connections to Apéry-like numbers, thereby generalizing the Apéry-Beukers framework beyond $\zeta(2)$. The approach leverages unitary equivalences between quantum models and holomorphic representations on Fock space and the unit disk, enabling explicit, computable expressions and revealing modular-like structures in the resulting Apéry-like components. The results illuminate deep links between spectral zeta values of quantum models and classical number-theoretic objects, with potential implications for irrationality questions and modularity phenomena in Apéry-type sequences.

Abstract

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.

Theorems & Definitions (51)

• Theorem 1.1: Corollary \ref{['cor_zeta_sum']}
• Theorem 1.2
• Corollary 1.3
• Lemma 2.1
• proof
• Theorem 2.2
• proof
• Example 2.3
• Remark 2.4
• Theorem 2.5