Generalized Rabi models: diagonalization in the spin subspace and differential operators of Dunkl type

A discrete parity $\mathbb{Z}_2$ symmetry of a two parameter extension of the quantum Rabi model which smoothly interpolates between the latter and the Jaynes-Cummings model, and of the two-photon and the two-mode quantum Rabi models enables their diagonalization in the spin subspace. A more general statement is that the respective sets of $2\times 2$ hermitian operators of the Fulton-Gouterman type and those diagonal in the spin subspace are unitary equivalent. The diagonalized representation makes it transparent that any question about integrability and solvability can be addressed only at the level of ordinary differential operators of Dunkl type. Braak's definition of integrability is shown (i) to contradict earlier numerical studies and (ii) to imply that any physically reasonable differential operator of Fulton-Gouterman type is integrable.