  In this paper we present the notion of globally integrable quantum system
that we introduced in [BL22]: we motivate it using the spectral theory of
pseudodifferential operators and then we give some results on linear and
nonlinear perturbations of a globally integrable quantum system. In particular,
we give a spectral result ensuring stability of most of its eigenvalues under
relatively bounded perturbations, and two results controlling the growth of
Sobolev norms when it is subject either to linear unbounded time dependent
perturbations or a small nonlinear Hamiltonian nonlinear perturbation.
