Mathematics

In this talk I will present a recent result on a class of linear time dependent Schrödingerequations on arbitrary at tori. In particular, a $|t|^{\epsilon}$ upper bound for any $\epsilon > 0$ on the growth of Sobolev norms of all the solutions will be given. As a main novelty, this result enables to deal with unbounded perturbations of the Laplacian, thus covering for instance the case of a particle moving in a time dependent electromagnetic field.The proof is based on a normal form technique and exploits ideas coming from classical dynamical systems: in particular, it is obtained as a quantum version of the proof of the classical Nekhoroshev theorem.Work in collaboration with D. Bambusi and R. Montalto.