Globally Integrable Quantum Systems and Their Perturbations

In this paper we present the notion of globally integrable quantum system that we introduced in [18]: 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 a linear unbounded time dependent perturbation or to a small nonlinear Hamiltonian nonlinear perturbation.