Growth of Sobolev norms in quasi integrable quantum systems

We prove an abstract result giving a \(\langle t \rangle^\varepsilon\) upper bound on the growth of the Sobolev norms of a time-dependent Schr\"odinger equation of the form \({i} \dot \psi = H_0 \psi + V (t)\psi\). Here \(H_0\) is assumed to be the Hamiltonian of a steep quantum integrable system and to be a pseudodifferential operator of order \({\tt d} > 1\); \(V (t)\) is a time-dependent family of pseudodifferential operators, unbounded, but of order \({\tt b} < {\tt d}\). 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.