Heat kernel for the quantum Rabi model: II. Propagators and spectral determinants

The quantum Rabi model (QRM) is widely recognized as an important in systems, particularly optics. Hamiltonian $H_{\text{Rabi}}$ known to have a parity decomposition $H_{\text{Rabi}} = H_{+} \oplus H_{-}$. In this paper, we give the explicit formulas for propagator of Schr\"odinger equation (integral kernel time evolution operator) and $H_{\pm}$ by Wick rotation (meromorphic continuation) corresponding heat kernels. addition, case full QRM, show that Hamiltonians $H_{\pm}$, spectral determinant is, up non-vanishing entire function, equal Braak $G$-function (for each parity) used prove integrability QRM. To do this, meromorphic continuation zeta function some its basic properties.