Symmetry in Nonlinear Mathematical Physics - 2009

Sergiy Dolya (B. Verkin Institute for Low Temperature Physics and Engineering, Kharkov, Ukraine)

Quasi-exactly solvable models, quadratic Lie algebras and special functions

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. These findings are used in studies of quasi-exactly solvable (QES) models in quantum mechanics. In particular, we show that the known two-photon Rabi Hamiltonian becomes QES at certain values of parameters. In doing so, there are two underlying Lie algebra, both of them being quadratic.