Growth of Sobolev norms in quasi-integrable quantum systems

We prove an abstract result giving a ⟨t⟩E upper bound on the growth of the Sobolev norms of a time-dependent Schrödinger equation of the form iψ ̇=H0ψ+V(t)ψ. {Here} H0 is assumed to be the Hamiltonian of a steep quantum integrable system and to be a {pseudodifferential} operator of order d>1 ; V(t) is a time-dependent family of pseudodifferential operators, unbounded, but of order b