Quasi-exactly solvable quartic Bose Hamiltonians

We consider Hamiltonians, which are even polynomials of the fourth order with respect to Bose operators. We find subspaces, preserved by the action of the Hamiltonian. These subspaces, being finite dimensional, include, nonetheless, states with an infinite number of quasi-particles, corresponding to the original Bose operators. The basis functions look rather simple in the coherent state representation and are expressed in terms of the degenerate hypergeometric function with respect to the complex variable labelling the representation. In some particular degenerate cases they turn (up to the power factor) into trigonometric or hyperbolic functions, Bessel functions or combinations of the exponent and Hermite polynomials. We find explicitly the relationship between coefficients at different powers of Bose operators that ensure quasiexact solvability of Hamiltonians.