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

The non-commutative harmonic oscillators Q α β NH with parameters α, β > 0, αβ > 1 and two-photon quantum Rabi models H △ , g 2 p with △ ≥ 0, | g | < 1 2 are both extensions of the one-dimensional harmonic oscillator. In the special case where α = β and △ = 0, it is immediately seen that Q α α NH is unitarily equivalent to α 2 − 1 1 − 4 g 2 H 0 , g 2 p . The purpose of this paper is to establish relationships between Q α β NH and H △ , g 2 p for the general cases △ > 0 and α ≠ β, and to show the fiber decomposition of Q α β NH in terms of H △ , g 2 p . We also construct Feynman-Kac formulas for e − t H △ , g 2 p and e − t Q α β NH . It is then considered the asymptotic behaviors of the spectral zeta function ζ_2p(s) of H △ , g 2 p .