Three-term asymptotic formula for large eigenvalues of the two-photon quantum Rabi model

Abstract

We prove that the spectrum of the two-photon quantum Rabi Hamiltonian consists of two eigenvalue sequences $(E_m^{+}{)}_{m=0}^{\infty }$, $(E_m^{-}{)}_{m=0}^{\infty }$ satisfying a three-term asymptotic formula with the remainder estimate $O(m^{-1}\ln m)$ when $m$ tends to infinity. Our asymptotic formula can be written so that the third term is given by an explicit oscillatory integral and an explicit remainder estimate.