Title: Abstract scattering theory and wave operators
Language: English
Authors: Barjon, Florent 
Qualification level: Diploma
Keywords: Streutheorie; Wellenoperatoren; Streuoperatoren
Scattering theory; Wave operators; Scattering operators
Advisor: Woracek, Harald
Issue Date: 2009
Number of Pages: 78
Qualification level: Diploma
Scattering theory has two sides. One is the direct investigation of scattering for Schrödinger equations intensively developed in quantum physics. The other side is called abstract scattering theory and is developed in an operator-theoretic framework. In this frame the Schrödinger operator is replaced by an arbitrary selfadjoint operator. Abstract scattering methods rely mainly on the concept of wave operators and scattering operators. The first objective of the present work is to give a systematic construction of abstract scattering by presenting conditions for the existence and completeness of wave operators. Once this is achieved, our aim is to obtain explicit formula for the scattering operator in the general case, similar to the one used in quantum mechanics.
We have divided this thesis in five chapters. After a brief introduction dedicated to physical motivations and definition of wave and scattering operators, we present in a second chapter preliminary facts about spectral theory and perturbation theory. The third chapter is dedicated to the study of abstract scattering in a single Hilbert space. There, we introduce simple wave operators and discuss their properties which we illustrate via practical examples taken from quantum scattering. In the fourth chapter, we set the stage for the extension of wave operators to scattering on two Hilbert spaces. In this view, we introduce identification operators in a very natural and physically relevant way known as algebraic scattering. The last chapter constitutes the main part of this thesis and is devoted to the study of wave operators in the frame of two-space scattering. After some general considerations, we focus on the investigation of existence and completeness of wave operators. We develop first time-dependent methods in which we deal essentially with time limits before turning to stationary techniques. Thanks to stationary representations, we derive in the very last part explicit formulas for scattering operators and scattering matrix.
URI: https://resolver.obvsg.at/urn:nbn:at:at-ubtuw:1-34231
Library ID: AC07452674
Organisation: E101 - Institut für Analysis und Scientific Computing 
Publication Type: Thesis
Appears in Collections:Thesis

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