Аннотация:
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.