Exactly integrable quantum dynamical systems

Abstract

In this chapter we study the quantum systems described by equations (3.2.8). On the quantum level the dynamical systems can be considered in different representations, in particular in the Schrödinger or Heisenberg ones. For the systems under study in the one-dimensional case we can solve the problem in both these representations. In the Schrödinger representation the wave functions are the matrix elements of the principal continuous series of unitary representations of noncompact real forms of complex semisimple Lie groups taken between the states with definite quantum numbers (generalized Whittaker vectors). At the same time the existence of a Hamiltonian formalism for the considered systems (5.8.1) enables us to apply the usual methods of perturbation theory as for the classical case, cf. 5.8. In such an approach the first term in the Hamiltonian (3.2.14) plays the role of the free Hamiltonian, whereas the second one with a factor λ describes the interaction in the system with the coupling constant λ. In complete analogy with the classical consideration the series of the perturbation theory are polynomials in λ and reproduce an exact solution of the corresponding system. In the one-dimensional case our constructions are essentially based on the representation theory of Lie groups and algebras and the final results are formulated completely in terms of this theory.