## A unified treatment of polynomial sectors of the Rabi models. (arXiv:1712.09371v1 [quant-ph])

The (driven) Rabi model, together with its two-mode, two-photon, and
asymmetric generalizations, are exotic examples of quasi-exactly solvable
models in that a corresponding 2nd order ordinary differential equation (ODE)
${\cal L}\psi=0$ with polynomial coefficients (i) is not Fuchsian one and (ii)
the differential operator ${\cal L}$ comprises energy E dependent terms $\sim
Ez d_z$, $Ez$, $E^2$. When recast into a Schr\"odinger equation (SE) form with
the first derivative term being eliminated and the coefficient of $d_x^2$ set
to one, such an equation is characterized by a nontrivially energy dependent
potential. The concept of a gradation slicing is introduced to analyze
polynomial solutions of such equations. It is shown that the ODE of all the
above Rabi models are characterized by the same unique set of grading
parameters. General necessary and sufficient conditions for the existence of a
polynomial solution are formulated. Unlike standard eigenvalue problems, the
condition that查看全文