Scattering Theory

Pure and Applied Mathematics, Vol. 26

Peter D. Lax, Ralph S. Phillips, Paul A. Smith, Samuel Eilenberg

• Publisher: Academic Press
• Year: 2016
• Print ISBN:
  9781483198606
• eBook ISBN:
  9781483223636
• Topic: Mathematics
• Subtopic: Probability & Statistics
• Index: Mathematics

Scattering Theory describes classical scattering theory in contrast to quantum mechanical scattering theory. The book discusses the formulation of the scattering theory in terms of the representation theory. The text also explains the relation between the behavior of the solution of the perturbed problem at small distances for large positive times and the analytic continuation of the scattering matrix. To prove the representation theorem, the text cites the methods used by Masani and Robertson in their work dealing with stationary stochastic processes. The book also applies the translation representation theory to a wave equation to obtain a comparison of the asymptotic properties of the free space solution with those of the solution in an exterior domain. The text discusses the solution of the wave equation in an exterior domain by fitting this problem into the abstract framework to get a verification of the hypotheses in some other theorems. The general theory of scattering can be applied to symmetric hyperbolic systems in which all sound speeds are different from zero, as well as to the acoustic equation which has a potential that can cause an energy form to become indefinite. The book is suitable for proponents of analytical mathematics, particle physics, and quantum physics.

Table of contents

• Front Cover
• Scattering Theory
• Copyright Page
• Table of Contents
• Dedication
• Preface
• Chapter 1. Introduction
• Chapter 2. Representation Theory and the Scattering Operator
• Chapter 3. A Semigroup of Operators Related to the Scattering Matrix
• Chapter 4. The Translation Representation for the Solution of the Wave Equation in Free Space
• Chapter 5. The Solution of the Wave Equation in an Exterior Domain
• Chapter 6. Symmetric Hyperbolic Systems, the Acoustic Equation with an Indefinite Energy Form, and the Schrödinger Equation
• APPENDIX 1: Semigroups of Operators
• APPENDIX 2: Energy Decay
• APPENDIX 3: Energy Decay for Star-Shaped Obstacles
• APPENDIX 4: Scattering Theory for Maxwell's Equations
• References
• Index