Isozaki, Hiroshi; Kitada, Hitoshi A remark on the micro-local resolvent estimates for two body Schrödinger operators. (English) Zbl 0611.35090 Publ. Res. Inst. Math. Sci. 21, 889-910 (1985). The resolvent \(R(z)=(-\Delta +V(x)-z)^{-1}\), where \(V(x)\in C^{\infty}({\mathbb{R}}^ n)\) (n\(\geq 2)\) and \[ \partial_ x^{\alpha}V(x)=O(| x|^{-| \alpha | -\epsilon_ 0})\quad as\quad | x| \to \infty \] for a constant \(0<\epsilon <1\) (\(\alpha\) multi-index, \(| \alpha | =\alpha_ 1+...+\alpha_ n)\), and the pseudo-differential operators \(P_{\pm}\) with symbols \(p_{\pm}(x,\xi)\), which fulfill the properties 1. \(| \partial_ x^{\alpha} \partial^{\beta}_{\xi} p_{\pm}(x,\xi)| \leq C_{\alpha \beta}(1+| x|^ 2)^{- | \alpha | /2},\) 2. \(p_{\pm}(x,\xi)=0\), if \(| x| <\epsilon\) or \(| \xi | <\epsilon\) for a constant \(\epsilon >0\) and 3. for constants \(-1<\mu_{\pm}<1\), \(p_+(x,\xi)=0\) if \(x\xi /| x| | \xi | <\mu_+\) and \(p_-(x,\xi)=0\) if \(x\xi /| x| | \xi | >\mu_-\) are considered. Then the following microlocal estimates for \(\lambda >a_ 0>0\) are derived: 1. For any \(2\geq 0\) and \(\delta >1\) \[ \| (1+| x|^ 2)^{s/2} P_{\mp}R(\lambda \pm i0)(1+| x|^ 2)^{- (s+\delta)/2}\| \leq C/\sqrt{\lambda}. \] 2. If \(\mu_+>\mu_-\), then for any \(s>0\) \[ \| (1+| x|^ 2)^{s/2} P_{\mp}R(\lambda \pm i0)P_{\pm}(1+| x|^ 2)^{s/2}\| \leq C/\sqrt{\lambda}. \] 3. If \(\mu_+>\mu_-\), then for any \(s>0\) \[ \| (1+| x|^ 2)^{s/2} P_-(R(\lambda +i0)-R_ j(\lambda - i0))P_+(1+| x|^ 2)^{s/2}\| \leq Cj^{-\epsilon_ 0}/\sqrt{\lambda}, \] if \(\chi\) is a \(C^{\infty}\)-function with \(\chi (x)=1\) for \(| x| <1\) and \(\chi (x)=0\) for \(| x| >2\) and \(R_ j(z)=(-\Delta +\chi (x/j)V(x)-z)^{-1}\). Reviewer: R.Weikard Cited in 17 Documents MSC: 35S05 Pseudodifferential operators as generalizations of partial differential operators 35A27 Microlocal methods and methods of sheaf theory and homological algebra applied to PDEs 35J10 Schrödinger operator, Schrödinger equation Keywords:micro-local resolvent; Schrödinger operator; S-matrix; pseudo- differential operators; symbols; microlocal estimates PDFBibTeX XMLCite \textit{H. Isozaki} and \textit{H. Kitada}, Publ. Res. Inst. Math. Sci. 21, 889--910 (1985; Zbl 0611.35090) Full Text: DOI References: [1] Isozaki, H. and Kitada, H., Micro-local resolvent estimates for two-body Schroinger operators, /. Functional Anal. 57 (1984), 270-300. · Zbl 0568.35022 · doi:10.1016/0022-1236(84)90104-6 [2] , Modified wave operators with time-independent modifiers, /. Fac. Sci. Tokyo Univ. Section I A, 1985, to appear. H 3 ] f Scattering matrices for two-body Schrodinger operators, preprint. [3] Kumanogo, H., Pseudo Differential Operators, The MIT Press, Cambridge, Mas- sachusets and London, England (1983). [4] 1 A calculus of Fourier integral operators on Rn and the fundamental solution for an operator of hyperbolic type, Comm in P.D.E., 1 (1976), 1-44. · Zbl 0331.42012 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.