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Multiple commutator estimates and resolvent smoothness in quantum scattering theory. (English) Zbl 0561.47007
Let H be a selfadjoint operator with domain \({\mathcal D}(H)\) in a Hilbert space \({\mathcal H}\). H is said to admit A as a conjugate operator at \(E\in R\), if A is a selfadjoint operator satisfying certain domain conditions, and if, furthermore, the following operator estimate holds: \[ (*)\quad E_ H(J)i[H,A]E_ H(J)\geq \alpha E_ H(J)+K. \] Here \(E_ H\) denotes the spectral measure of H, \(J=(E-\delta\), \(E+\delta)\), \(\delta >0\), \(\alpha >0\), and K is compact. One furthermore assumes that the multiple commutators \(i[i[H,A],A]...\) up to order n define bounded operators from \({\mathcal D}(H)\) (graph norm) to \({\mathcal H}\). A weaker condition is also needed on the \((n+1)\)-fold commutator.
The main results are that the boundary values \(R(\alpha \pm i0)=\lim_{\epsilon \downarrow 0}(H-\lambda \mp i\epsilon)^{-1}\) exist and are \(C^ n\) w.r.t. \(\lambda\) in suitable spaces defined using A. Here \(\lambda \in J\setminus \sigma_ p(H),\) where \(\sigma_ p(H)\) denotes the point spectrum of H. Under condition (*) \(\sigma_ p(H)\) is known to be discrete in J. Detailed results are too complicated to be stated here.
The applications include an abstract scattering theory using a conjugate operator, and a proof of completeness of modified wave operators for certain Schrödinger operators with long range potentials.

MSC:
47A40 Scattering theory of linear operators
47A10 Spectrum, resolvent
81U05 \(2\)-body potential quantum scattering theory
47F05 General theory of partial differential operators
47B47 Commutators, derivations, elementary operators, etc.
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