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We study the solvability of the time-independent matrix Schr\"odingerdifferential equations of the quantum Rabi model and its 2-photon and two-modegeneralizations in Bargmann Hilbert spaces of entire functions. We show thatthe Rabi model and its 2-photon and two-mode analogs are quasi-exactlysolvable. We derive the exact, closed-form expressions for the energies and theallowed model parameters for all the three cases in the solvable subspaces. Upto a normalization factor, the eigenfunctions for these models are given bypolynomials whose roots are determined by systems of algebraic equations.

On the solvability of the quantum Rabi model and its 2-photon and two-mode generalizations