Spectral zeta function and ground state of quantum Rabi model

The spectral zeta function ζ_g(s;g²+τ)=∑_n=0^∞[Formula presented] of the quantum Rabi Hamiltonian is considered, where τ>0 and E_n(g) denotes nth eigenvalue of the quantum Rabi Hamiltonian H. Let ζ(s;τ)=∑_n=0^∞[Formula presented] be the Hurwitz zeta function. It is shown that lim_|g|→∞⁡ζ_g(s;g²+τ)=2ζ(s;τ). Moreover the path measure Π_∞ associated with the ground state of H is constructed on a discontinuous path space, and several applications are shown.