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Micro-local resolvent estimates for 2-body Schrödinger operators. (English) Zbl 0568.35022
The aim of the present paper is to obtain micro-local resolvent estimates for the Schrödinger operator $$H=-\Delta +V$$ on $$L^ 2({\mathbb{R}}^ n)$$, where V is a real $$C^{\infty}$$-function on $${\mathbb{R}}^ n$$ such that for some $$\epsilon_ 0>0$$ $(1)\quad \partial^{\alpha}V(x)=O(| x|^{-| \alpha | -\epsilon_ 0})\quad as\quad | x| \to \infty$ for all multi-index $$\alpha$$. By micro-local resolvent estimates the authors mean estimates for the operators $$P_{\mp}R(\lambda \pm iO),P_{\mp}R(\lambda \pm iO)P_{\pm}$$, etc., in a weighted $$L^ 2$$-space $$L^{2,s}$$. Here $$R(\lambda \pm iO)$$ stands for the strong limit $$-\lim_{\epsilon \downarrow 0}(H- \lambda_{\mp}i\epsilon)^{-1}$$ in $$B(L^{2,\gamma},L^{2,-\gamma})$$ $$(\gamma >)$$, which exists due to the limiting absorption principle, and $$P_{\pm}$$ are the pseudo-differential operators defined by $P_{\pm}f(x)=\int_{{\mathbb{R}}^ n}\int_{{\mathbb{R}}^ n}e^{i(x-y)\cdot \xi}p_{\pm}(x,\xi)dy d\xi.$ Of course $$p_{\pm}$$ has to satisfy certain conditions. The results of this paper can be summarized as follows:
Theorem 1. If $$a_ 0>0$$ and $$s>$$ then $\| P_{\mp}R(\lambda \pm iO)f\|_{s-1}\leq C\lambda^{-1/2}\| f\|_ s\quad for\quad any\quad \lambda >a_ 0.$ Theorem 2. If $$\inf_{x\in {\mathbb{R}}^ n}dist(\sup p p_+(x,\cdot)$$, supp $$p_-(x,\cdot))>0$$ then $(i)\quad \| P_{\mp}R(\lambda \pm iO)P_{\pm}f\|_ M\leq (c/\sqrt{\lambda})\| f\|_{-N},$ $(ii)\quad \| P_{\mp}(R(\lambda \pm iO)-R_ j(\lambda \pm iO))P_{\pm}f\|_ M\leq (c/\lambda)j^{-\epsilon_ 0/2}\| f\|_{-N}$ for any $$M,N>0$$ and $$\lambda >a_ 0>0.$$
Notice that the constant c in (ii) is independent of j and that $$R_ j(z)=(-\Delta +\chi (\cdot /j)V-z)^{-1},$$ where $$\chi$$ is a real-valued $$C^{\infty}$$-function satisfying $$\chi (x)=1$$ for $$| x| <1$$ and $$\chi (x)=0$$ for $$| x| >2$$. The micro-local estimates established in this paper have applications in the study of the scattering matrix. In a forthcoming paper the authors announce the discussion of problems of regularities and singularities of scattering amplitudes and also a reconstruction formula of the potential.
Reviewer: H.Leinfelder

##### MSC:
 35J10 Schrödinger operator, Schrödinger equation 35P25 Scattering theory for PDEs
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##### References:
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