Special values of spectral zeta functions of one-, two-photon quantum Rabi models and non-commutative harmonic oscillators
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| Format: | Preprint |
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2026
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| _version_ | 1866911416097701888 |
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| author | Nakahama, Ryosuke |
| author_facet | Nakahama, Ryosuke |
| contents | 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. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2602_01628 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Special values of spectral zeta functions of one-, two-photon quantum Rabi models and non-commutative harmonic oscillators Nakahama, Ryosuke Mathematical Physics Number Theory 81Q10, 11M41 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. |
| title | Special values of spectral zeta functions of one-, two-photon quantum Rabi models and non-commutative harmonic oscillators |
| topic | Mathematical Physics Number Theory 81Q10, 11M41 |
| url | https://arxiv.org/abs/2602.01628 |