# Scattering Theory

## 1st Edition

### Pure and Applied Mathematics, Vol. 26

**Authors:**Peter D. Lax Ralph S. Phillips

**Editors:**Paul A. Smith Samuel Eilenberg

**eBook ISBN:**9781483223636

**Imprint:**Academic Press

**Published Date:**1st January 1967

**Page Count:**288

## Description

*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

Preface

Chapter I. Introduction

1. The Dynamic Approach

2. Scattering Theory Formulated in Terms of Representation Theory

3. A Semigroup of Operators Related to the Scattering Matrix

4. The Form of the Scattering Matrix

5. A Simple Example

6. Scattering Theory for Transport Phenomena

7. Notes and Remarks

Chapter II. Representation Theory and the Scattering Operator

1. The Discrete Case

2. The Scattering Operator in the Discrete Case

3. The Continuous Case

4. The Scattering Operator in the Continuous Case

5. Notes and Remarks

Chapter III. A Semigroup of Operators Related to the Scattering Matrix

1. The Related Semigroups

2. On Semigroups of Contraction Operators

3. Spectral Theory

4. A Spectral Mapping Theorem

5. Applications of the Spectral Theory

6. Equivalent Incoming and Outgoing Representations

7. Notes and Remarks

Chapter IV. The Translation Representation for the Solution of the Wave Equation in Free Space

1. The Hilbert Space H0 and the Group {U0(t)}

2. Spectral and Translation Representations of {U0(t)}

3. The Operator # Extended to Distributions

4. Translation Representation for Outgoing and Incoming Data with Infinite Energy

5. Notes and Remarks

Chapter V. The Solution of the Wave Equation in an Exterior Domain

1. The Hilbert Space H and the Group {U(t)}

2 . Energy Decay and Translation Representations

3 . The Semigroup {Z(t)}

4 . The Relation between the Semigroup {Z(t) ) and the Solutions of the Reduced Wave Equation

5 . The Scattering Matrix

6 . Notes and Remarks

Chapter VI. Symmetric Hyperbolic Systems, the Acoustic Equation with an Indefinite Energy Form, and the Schrödinger Equation

Part 1. Symmetric Hyperbolic Systems

1. Translation Representation in Free Space

2. Solutions of Hyperbolic Systems in an Exterior Domain

Part 2. The Acoustic Equation with an Indefinite Energy Form and the Schrödinger Equation

3 . Scattering for the Acoustic Equation with an Indefinite Energy Form

4 . The Schrödinger Scattering Matrix

5 . Notes and Remarks

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

## Details

- No. of pages:
- 288

- Language:
- English

- Copyright:
- © Academic Press 1967

- Published:
- 1st January 1967

- Imprint:
- Academic Press

- eBook ISBN:
- 9781483223636

## About the Author

### Peter D. Lax

### Ralph S. Phillips

## About the Editor

### Paul A. Smith

### Samuel Eilenberg

### Affiliations and Expertise

Columbia University