Fiber decomposition of non-commutative harmonic oscillators by two-photon quantum Rabi models

TL;DR

The paper uncovers a precise fiber-decomposition linking the non-commutative harmonic oscillator and the two-photon quantum Rabi model, showing how NcHO can be expressed as a direct sum of 2pQRM fibers via a unitary transform. It builds new Feynman-Kac representations for the semigroups of both systems and analyzes the spectral zeta function of the 2pQRM through path measures, deriving limiting forms tied to Hurwitz zeta functions. A parallel fiber analysis for the one-photon case and a detailed study of the parity and symmetry structures are developed, together with perturbative concavity results for the lowest eigenvalues. The results illuminate structural and spectral correspondences between NcHO and 2pQRM, offering tools for rigorous spectral analysis and potential applications to quantum simulation and mathematical physics. The work extends the algebraic-analytic understanding of higher-order Rabi models by embedding them in the NcHO framework via explicit decompositions and probabilistic representations.

Abstract

The non-commutative harmonic oscillators (NcHO) and 2p-quantum Rabi models (2pQRM) are extensions of harmonic oscillators. The purpose of this paper is to give a relationship between NcHO and 2pQRM, and the fiber decomposition of NcHO by 2pQRM is shown. We also construct Feynman-Kac formulas of NcHO and 2pQRM. Then asymptotic behaviors of the spectral zeta function of 2pQRM is considered.

Figures (1)

• Figure 1: (1) of Lemma \ref{['qere']}

Theorems & Definitions (43)

• Proposition 2.1
• Lemma 2.2: spectrum of $H_{0,g}^{\rm 2p}$
• Lemma 2.3: self-adjointness of $H_{\triangle, g}^{\rm 2p}$
• Lemma 2.4: essential self-adjointness of $Q_{\alpha\beta}^{\rm NH}$
• Lemma 2.5: spectrum of $K_{\alpha\beta}^{\rm NH}$
• Lemma 2.6: self-adjointness of $Q_{\alpha\beta}^{\rm NH}$
• Remark 2.7
• Remark 2.8
• Lemma 2.9
• Theorem 2.10: concavity of eigenvalue curves of $H_{\triangle, g}^{\rm 2p}$