  The two-photon quantum Rabi model with quadratic coupling is studied using
extended squeezed states and we derive $G$-functions for Bargmann index $q=1/4$
and $3/4$. The simple singularity structure of the $G$-function allows to draw
conclusions about the distribution of eigenvalues along the real axis. The
previously found picture of the spectral collapse at critical coupling
$g_{\mathrm{c}}$ has to be modified regarding the low lying states, especially
the ground state: We obtain a finite gap between ground state and the continuum
of excited states at the collapse point. For large qubit splitting, also other
low lying states may be separated from the continuum at $g_{\mathrm{c}}$. We
have carried out a perturbative analysis allowing for explicit and simple
formulae of the eigenstates. Interestingly, a vanishing of the gap between
ground state and excited continuum at $g_{\mathrm{c}}$ is obtained in each
finite order of approximation. This demonstrates cleary the non-pertubative
nature of the excitation gap. We corroborate these findings with a variational
calculation for the ground state.
