Quasi-exactly solvable models based on special functions

We suggest a systematic method of extension of quasiexactly solvable (QES) systems. We construct finite-dimensional subspaces on the basis of special functions (hypergeometric, Airy, Bessel ones) invariant with respect to the action of differential operators of the second order with polynomial coefficients. As an example of physical applications, we show that the known two-photon Rabi Hamiltonian becomes quasiexactly solvable at certain values of parameters when it can be expressed in terms of corresponding QES operators related to the hypergeometric function. (C) 2008 American Institute of Physics.