On Integrability of Dynamical Systems

A classical dynamical system may have smooth integrals of motion and not have analytic ones; i.e., the integrability property depends on the category of smoothness. Recently it has been shown that any quantum dynamical system is completely integrable in the category of Hilbert spaces and, moreover, is unitarily equivalent to a set of classical harmonic oscillators. The same statement holds for classical dynamical systems in the Koopman formulation. Here we construct higher conservation laws in an explicit form for the Schrödinger equation in the multidimensional space under various fairly wide conditions on the potential.