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On time dependent Schrödinger equations: global well-posedness and growth of Sobolev norms

Speaker: 
Alberto Maspero
Institution: 
SISSA
Schedule: 
Tuesday, March 28, 2017 - 14:00
Location: 
A-133
Abstract: 

 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.

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