In this talk we consider time dependent Schrodinger linear PDEs of the form $\text{im } \partial_t \psi = (H +V(t)) \psi $ where $V(t)$ is a perturbation smooth in time and $H$ is a self-adjoint positive operator whose spectrum can be enclosed in spectral clusters whose distance is increasing. We prove that the Sobolev norms of the solution grow at most as $t^\epsilon$ when $t\mapsto \infty$, for any $\epsilon >0$. If $V(t)$ is analytic in time we improve the bound to $(\log t)^\gamma$, for some $\gamma >0$. The proof follows the strategy of adiabatic approximation of the flow. We recover most of known results and obtain new estimates for several models including $1$-degree of freedom Schrodinger operators on $\mathbb R$. This is a joint work with Didier Robert.
On time dependent Schrödinger equations: global well-posedness and growth of Sobolev norms
Research Group:
Speaker:
Alberto Maspero
Institution:
SISSA
Schedule:
Tuesday, March 28, 2017 - 14:00
Location:
A-133
Abstract: