Spectral Determinant of the Two-Photon Quantum Rabi Model

The various generalized spectral determinants (G-functions) of the two-photon quantum Rabi model are analyzed with emphasis on the qualitative aspects of the regular spectrum. Whereas all of them yield at least a subset of the exact regular eigenvalues, only the G-function proposed by Chen et al. in 2012 exhibits an explicitly known pole structure which dictates the approach to the collapse point. This function is derived rigorously employing the Z4${\mathbb{Z}}_{4}$-symmetry of the model and shown that its zeros correspond to the complete regular spectrum.