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Birkhäuser Basel eBooks

Moving Potentials and the Completeness of Wave Operators. Part II: Propagating Observables on Scattering States

Hagen Neidhardt

Published Jan 1, 1990• 2 citations
Publisher

Abstract

The paper is devoted to the time evolution of the position and impulse operators as well as certain functions of them, for example the dilation and classical kinetic energy operator, in quantum mechanical scattering theory with time-dependent and travelling along fixed trajectories long range potentials. These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

DOI
10.1007/978-3-0348-7306-2_12

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Birkhäuser Boston eBooks•Jan 1, 1990•1 citation
The paper is devoted to the time evolution of the position and impulse operators as well as certain functions of them, for example the dilation and classical kinetic energy operator, in quantum mechanical scattering theory with time-dependent and travelling along fixed trajectories long range potentials. These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.
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Scattering theory deals with the evolution of states of a system and, in particular, attempts to relate the state of a system as t → +∞ with the state of that system as t → −∞. In Chapter 3 we indicated that such a comparison could be made in terms of so-called wave operators. In this chapter we shall introduce these quantities rather more properly and outline some of their more important properties. This we can do most conveniently in an abstract setting; we shall consider specific cases later.
The aim of the lecture is to briefly describe the mathematical background of scattering theory for two- and three-particle quantum systems. We discuss basic objects of the theory: wave and scattering operators and the corresponding scattering matrix and illustrate them on the example of the Schrödinger equation. Our goal is to present time-dependent and stationary approaches and to describe the underlying mathematical methods. We also give a sketch of scattering theory for three interacting quantum particles including a difficult problem of the asymptotic completeness of scattering channels. Along with traditional results, we discuss new scattering channels arising for long-range pair interactions.