Quantum Inozemtsev model , quasi - exact solvability and N - fold supersymmetry

Inozemtsev models are classically integrable multi-particle dynamical systems related to Calogero-Moser models. Because of the additional q6 (rational models) or sin 2q (trigonometric models) potentials, their quantum versions are not exactly solvable in contrast with Calogero-Moser models. We show that quantum Inozemtsev models can be deformed to be a widest class of partly solvable (or quasi-exactly solvable) multi-particle dynamical systems. They posses N -fold supersymmetry which is equivalent to quasi-exact solvability. A new method for identifying and solving quasiexactly solvable systems, the method of pre-superpotential, is presented.