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

We prove that the spectrum of the two-photon quantum Rabi Hamiltonian consists of two eigenvalue sequences (Eₘ⁺)ₘ₌₀∞ , (Eₘ⁻)ₘ₌₀∞ satisfying a three-term asymptotic formula with the remainder estimate O(m⁻¹ln 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.