Growth of Sobolev norms in quasi-integrable quantum systems

We prove an abstract result giving a t^\epsilon upper bound on the growth of the Sobolev norms of a time-dependent Schrodinger equation of the form i \psi_t = H_0 \psi +V(t) \psi. Here H_0 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 < d. The abstract theorem is then applied to perturbations of the quantum anharmonic oscillators in dimension 2 and to perturbations of the Laplacian on a manifold with integrable geodesic flow, and in particular Zoll manifolds, rotation-invariant surfaces and Lie groups. The proof is based on a quantum version of the proof of the classical Nekhoroshev theorem.