The discrete spectrum states of finite-dimensional quantum systems connected with Lie algebra

The problem of finding the energy levels and wave functions for discrete spectrum states of a class of Hamiltonians connected with root systems of the Lie algebras is considered. In a limiting case these Hamiltonians describe the motion of quantum interacting particle systems on a line in the Morse oscillator. It is shown that this problem can be reduced to algebraic one; the exact expressions determining the dependence of energy levels on the parameters of the Hamiltonian are obtained