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

The non-commutative harmonic oscillators QαβNH with parameters α, β > 0, αβ > 1 and two-photon quantum Rabi models H△,g2p with △ ≥ 0, |g|<12 are both extensions of the one-dimensional harmonic oscillator. In the special case where α = β and △ = 0, it is immediately seen that QααNH is unitarily equivalent to α2−11−4g2H0,g2p. The purpose of this paper is to establish relationships between QαβNH and H△,g2p for the general cases △ > 0 and α ≠ β, and to show the fiber decomposition of QαβNH in terms of H△,g2p. We also construct Feynman–Kac formulas for e−tH△,g2p and e−tQαβNH. It is then considered the asymptotic behaviors of the spectral zeta function ζ2p(s) of H△,g2p.