Growth of Sobolev norms for unbounded perturbations of the Laplacian on flat tori (towards a quantum Nekhoroshev theorem)

I will present a study of the time dependent Schrödinger equation
$$ -i\psi_t=-\Delta\psi+{\cal V}(t,x,-i\nabla)\psi $$
on a flat $d$ dimensional torus. Here ${\cal V}$ is a time dependent pseudodifferential operator of order strictly smaller than 2. The main result I will give is an estimate ensuring that the Sobolev norms of the solutions are bounded by $t^{\epsilon}$. The proof is a quantization of the proof of the Nekhoroshev theorem, both analytic and geometric parts.
Previous results of this kind were limited either to the case of bounded perturbations of the Laplacian or to quantization of systems with a trivial geometry of the resonances, lik harmonic oscillators or 1-d systems.
In this seminar I will present the result and the main ideas of the proof.
This is a joint work with Beatrice Langella and Riccardo Montalto.