It is shown that any quantum dynamical system (Schrodinger equation) is completely integrable in the category of Hilbert spaces and, moreover, unitarily equivalent to a system of noninteracting classical harmonic oscillators. The same statement holds for classical dynamical systems. Higher integrals of motion are presented. We make the formulation on integrability of any quantum dynamical systems sharper by using the Radon measure in the spectral theorem. We also construct the higher conservation laws in explicit form for the Schrodinger equation in multidimensional space under rather wide conditions on the potential using the stationary theory of scattering and expansion in eigenfunctions. Higher integrals of motion for multidimensional nonlinear Klein-Gordon and Schrodinger equation are constructed. A criterion for the occurrence of quantum chaos in integrable systems is presented. A quantum mechanical system with the finite-dimensional Hamiltonian with rational entries is integrable in radicals if the Galois group of its characteristic polynomial is solvable.
References
[1] Igor V. Volovich, Complete Integrability of Quantum and Classical Dynamical Systems, $p$-Adic Numbers, Ultrametric Analysis and Applications, 2019, Volume 11, Number 4, Page 328.
[2] Igor V. Volovich, Remarks on the complete integrability of quantum and classical dynamical systems, arXiv:1911.01335.
[3] I. V. Volovich, On Integrability of Dynamical Systems, Proceedings of the Steklov Institute of Mathematics, volume 310, pages 70-77 (2020).