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A remark on the micro-local resolvent estimates for two body Schrödinger operators. (English) Zbl 0611.35090

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

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
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[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
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