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Spectral and scattering theory by the conjugate operator method. (English) Zbl 0780.47010

The aim of this article is to describe the power of the conjugate operator method. That allows to study Hamiltonians outside the scope of other methods in spectral and scattering analysis. There are studied selfdjoint operators of the form \(H=h(-i\nabla)+V\). The function \(h\) covers for instance all hypoelliptic polynomials. The assumptions on \(V\) allow very general short and long range parts of the potential. A complete spectral analysis of \(H\) is possible by this conjugate operator method (absence of singularly continuous spectra, multiplicity of eigenvalues and their possible accumulation points, limiting absorption principle, existence and completeness of relative wave operators).
Some abstract tools for the theory are developed (real interpolation, Tauberian theorem, unitary groups in Friedrichs couples). The conjugate operator method is sketched. Applications for pseudo-differential operators are given. On account of its high pedagogical level the article is useful as an introduction into this kind of spectral analysis.
Reviewer: M.Demuth (Potsdam)

MSC:

47A40 Scattering theory of linear operators
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