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

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