Spectral zeta function and ground state of quantum Rabi model

Abstract

The spectral zeta function ${\zeta }_g(s;g^2+\tau )={\sum }_{n=0}^{\infty }\frac{1}{{\left(E_g\right(n)+g^2+\tau )}^s}$ of the quantum Rabi Hamiltonian is considered, where $\tau >0$ and $E_n\left(g\right)$ denotes nth eigenvalue of the quantum Rabi Hamiltonian H. Let $\zeta (s;\tau )={\sum }_{n=0}^{\infty }\frac{1}{{(n+\tau )}^s}$ be the Hurwitz zeta function. It is shown that ${\lim }_{\left|g\right|\to \infty }{\zeta }_g(s;g^2+\tau )=2\zeta (s;\tau )$. Moreover the path measure ${\Pi }_{\infty }$ associated with the ground state of H is constructed on a discontinuous path space, and several applications are shown.