Dynamical symmetry, integrability of quantum systems, and general character of quantum regular motion

The notion of quantum-classical correspondence is carefully investigated in order to prepare firm grounds for studying the spatiotemporal evolution of quantum states in the same spirit as for corresponding classical cases. Three relevant problems, (1) the integrability of dynamical equations of quantum systems, (2) the initial minimum uncertainty states one-to-one correspondent to classical phase points, and (3) the effective Planck constants for systems having analogous dynamical properties but exhibiting different quantum effects, have been successfully resolved. Then the solution ρ_γ(t) of the dynamical equation of a quantum integrable system is shown to be expressed as an analytical functional of the initial minimum uncertainty state ρ_γ⁰ varying smoothly with γ and t. Such a general character of the quantum regular motion serves as a reference for the study of quantum irregular motion under the action of perturbed Hamiltonian.