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 ∞<!-- ∞ --> E_n^+ ₙ₌₀∞ , { E n −<!-- − --> } n = 0 ∞<!-- ∞ --> E_n^- ₙ₌₀∞ , and their large n n asymptotic behavior with error term O ⁡<!-- ⁡ --> ( n −<!-- − --> 1 / 2 ) O(n-1/2) is described. The principal tool is the method of near-similarity of operators introduced by G. V. Rozenblum and developed in works of J. Janas, S. Naboko, and E. A. Yanovich (Tur).