Micro-local resolvent estimates for 2-body Schrödinger operators (Q1060343)

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.