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Geometric scattering theory. (English) Zbl 0849.58071
Stanford Lectures. Cambridge: Cambridge Univ. Press. xi, 116 p. (1995).
These are lecture notes on scattering theory of the Laplace and Schrödinger operator. First the author presents the scattering theory for the pair \((\Delta, \Delta + V), V \in C_c^\infty (\mathbb{R}^n)\) in Euclidean space \(\mathbb{R}^n\), i.e. scattering theory for potential perturbations of the flat Laplacian. Then he gives an exposition of inverse scattering, i.e. the determination of \(V\) from the scattering matrix and amplitude. Thereafter he studies the poles of the analytic continuation of the scattering matrix an their relation to the spectrum. This finishes potential perturbation. In a second part the author discusses obstacle scattering and more general Riemannian manifolds, in particular asymptotically flat metrics and cylindrical end metrics. This requires more substantial material like index theorems, trace formulas etc. In a final chapter, the author is concerned with warped product metrics and covers this case really very smart. The whole exposition is written quite unformal, contains all substantial arguments and can be warmly recommended to any reader interested in this topic.

MSC:
58J50 Spectral problems; spectral geometry; scattering theory on manifolds
58J20 Index theory and related fixed-point theorems on manifolds
58-02 Research exposition (monographs, survey articles) pertaining to global analysis
35P25 Scattering theory for PDEs
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