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Absolutely continuous and singular spectral shift functions. (English) Zbl 1276.47013

In the paper under review, the author develops a variant of the abstract scattering theory for trace class perturbations (as presented in the book [D. R. Yafaev, Mathematical scattering theory. General theory. Providence, RI: American Mathematical Society (AMS) (1992; Zbl 0761.47001)]) and uses it to prove that the singular part of the spectral shift function (SSF) is integer valued. This result is based on a new representation formula given for the scattering operator and provides a new proof of the Birman-Krein formula, that relates the determinant of the scattering matrix to the SSF.
The author essentially follows the usual abstract scattering theory in [loc. cit.] but with the help of a particular, flexible tool, a so-called “frame”. A frame-dependent spectral representation of the free operator is constructed at the very beginning and is used to defined wave matrices. The usual wave operators are recovered by integration of the latter. The advantage of this approach is in a better control of the set of full Lebesgue measure on which the objects and formulae are valid (this set actually depends only on the unperturbed operator and the frame). To get the main result and the new proof of the Birman-Krein formula, the author makes use of the Pushnitskij \(\mu\)-invariant. It would be interesting to weaken the trace class assumption on the perturbation (to relatively trace class perturbation for instance) to enlarge the possibilities of application of the theory.
We recommend the reading of the Introduction that provides a good overview of the theory. The text contains some small, unimportant mistakes (like the sentence following Definition 0.2.1) that could have been corrected by the referee (at least in the Introduction).

MSC:

47A40 Scattering theory of linear operators
47A70 (Generalized) eigenfunction expansions of linear operators; rigged Hilbert spaces
81U99 Quantum scattering theory

Citations:

Zbl 0761.47001
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