Behavior of large eigenvalues for the two-photon asymmetric quantum Rabi model

The asymptotic behavior of large eigenvalues is studied for the two-photon quantum Rabi model with a finite bias. It is proved that the spectrum of this Hamiltonian model consists of two eigenvalue sequences $\{E_n^+\}_{n=0}^{\infty}$, $\{E_n^-\}_{n=0}^{\infty}$, and their large $n$ asymptotic behavior with error term $\mathrm{O}(n^{-1/2})$ is described. The principal tool is the method of near-similarity of operators introduced by G. V. Rozenbljum and developed in works of J. Janas, S. Naboko, and E. A. Yanovich (Tur).