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Abstract: |
In this talk, I will first review some basic concepts and classical results in the linear scattering theory for the Schr?dinger operator $H=-\ Delta+V$ with a short range potential $V(x)$ , as well as the short range perturbation theory for the so-called simply characteristic operators established by Agmon and H?rmander in the 1970s where higher order Schr?dinger operators $H=(-\ Delta)^m+V(x,D)$ are included. Then I will introduce how to define a short range potential $V(x)$ and to establish the scattering theory for the fractional Schr?dinger operator $H=(-\ Delta)^{s/2}+V$ with full range $s>0$. When $s$ is not an even integer, the main difficulty is to deal with the non-local aspect of $(-\ Delta)^{s/2}$, while its resolvent estimates are crucial since the theory relies on the study of the spectral measure of $H$. Our results are sharp so far with respect to the decay assumption on $V$, and this is a joint-work with Rui Zhang and Prof. Quan Zheng. |
