Remarks on the complete integrability of quantum and classical dynamical systems

It is noted that the Schrodinger equation with any self-adjoint Hamiltonian is unitary equivalent to a set of non-interacting classical harmonic oscillators and in this sense any quantum dynamics is completely integrable. Higher order integrals of motion are presented. That does not mean that we can explicitly compute the time dependence for expectation value of any quantum observable. A similar result is indicated for classical dynamical systems in terms of Koopman's approach. Explicit transformations of quantum and classical dynamics to the free evolution by using direct methods of scattering theory and wave operators are considered. Examples from classical and quantum mechanics, and also from nonlinear partial differential equations and quantum field theory are discussed. Higher order integrals of motion for the multi-dimensional nonlinear Klein-Gordon and Schrodinger equations are mentioned.