The quantum anharmonic oscillator and quasi-exactly solvable Bose systems

We extend the notion of quasi-exactly solvable (QES) models from potential ones and differential equations to Bose systems. We obtain conditions under which algebraization of part of the spectrum occurs. In some particular cases simple exact expressions for several energy levels of an anharmonic Bose oscillator are obtained explicitly. The corresponding results do not exploit perturbation theory and include the strong-coupling regime. A generic Hamiltonian under discussion cannot, in contrast to QES potential models, be expressed as a polynomial in generators of sit algebra. The suggested approach is extendable to many-particle Bose systems with interaction.