Globally integrable quantum systems and their perturbations

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.