##
**Mathematical scattering theory. General theory.**
*(English)*
Zbl 0761.47001

Translations of Mathematical Monographs. 105. Providence, RI: American Mathematical Society (AMS). x, 341 p. (1992).

The book is mainly devoted to the stationary, i.e. time-independent, mathematical scattering theory. In the preliminaries several necessary facts needed for describing the scattering theory are explained, e.g. parts of measure theory, decomposition of spectra of selfadjoint operators, theory of compact operators, Fredholm theory. Then the time- dependent two-space scattering theory is introduced in a straightforward manner: existence, completeness and properties of wave and scattering operators, invariance principle, link to the time-independent approach.

However, the main objective in this book is the abstract stationary scattering theory. The Kato smoothness and the Friedrichs-Faddeev models are studied. The concept of weak smoothness gives explicit representations of scattering matrix elements. Moreover, the trace class and the local trace class theory are explained, including some special problems like double Stieltjes operator integrals.

This stationary theory is used to analyse a series of spectral properties for the scattering matrices and the spectral shift function. Concerning the scattering matrix there are considered: the corresponding invariance principle, the continuity in case of changing the potentials, behaviour for smooth perturbations, their spectra in the complex plane and the continuity of these spectra for different couplings, estimates for the scattering cross section. The spectral shift function is studied for pairs of selfadjoint or unitary operators. Relations to the scattering matrices are given.

The book fills a gap in the literature because it emphasizes the time- independent scattering theory. The explanations and descriptions are clear and understandable for readers who are familiar with the framework and fundamentals of the scattering theory. In the preface of every chapter its content is described exactly. The book provides a complete view of the time-independent theory excluding the resonance theory.

It contains no examples, although they would be helpful for some arguments and facts. Probably all the applications and the link of quantum mechanics will follow in the second volume. The list of references is selected and collects mainly the East-European literature.

However, the main objective in this book is the abstract stationary scattering theory. The Kato smoothness and the Friedrichs-Faddeev models are studied. The concept of weak smoothness gives explicit representations of scattering matrix elements. Moreover, the trace class and the local trace class theory are explained, including some special problems like double Stieltjes operator integrals.

This stationary theory is used to analyse a series of spectral properties for the scattering matrices and the spectral shift function. Concerning the scattering matrix there are considered: the corresponding invariance principle, the continuity in case of changing the potentials, behaviour for smooth perturbations, their spectra in the complex plane and the continuity of these spectra for different couplings, estimates for the scattering cross section. The spectral shift function is studied for pairs of selfadjoint or unitary operators. Relations to the scattering matrices are given.

The book fills a gap in the literature because it emphasizes the time- independent scattering theory. The explanations and descriptions are clear and understandable for readers who are familiar with the framework and fundamentals of the scattering theory. In the preface of every chapter its content is described exactly. The book provides a complete view of the time-independent theory excluding the resonance theory.

It contains no examples, although they would be helpful for some arguments and facts. Probably all the applications and the link of quantum mechanics will follow in the second volume. The list of references is selected and collects mainly the East-European literature.

Reviewer: M.Demuth (Potsdam)

### MSC:

47A40 | Scattering theory of linear operators |

81U20 | \(S\)-matrix theory, etc. in quantum theory |

47-02 | Research exposition (monographs, survey articles) pertaining to operator theory |