Scattering Theory for Transport Phenomena

✍ Scribed by Hassan Emamirad

• Publisher: Springer
• Year: 2021
• Tongue: English
• Leaves: 197
• Series: Mathematical Physics Studies
• Category: Library

✦ Synopsis

The scattering theory for transport phenomena was initiated by P. Lax and R. Phillips in 1967. Since then, great progress has been made in the field and the work has been ongoing for more than half a century. This book shows part of that progress. 

The book is divided into 7 chapters, the first of which deals with preliminaries of the theory of semigroups and C*-algebra, different types of semigroups, Schatten–von Neuman classes of operators, and facts about ultraweak operator topology, with examples using wavelet theory.

Chapter 2 goes into abstract scattering theory in a general Banach space. The wave and scattering operators and their basic properties are defined. Some abstract methods such as smooth perturbation and the limiting absorption principle are also presented. Chapter 3 is devoted to the transport or linearized Boltzmann equation, and in Chapter 4 the Lax and Phillips formalism is introduced in scattering theory for the transport equation. 

In their seminal book, Lax and Phillips introduced the incoming and outgoing subspaces, which verify their representation theorem for a dissipative hyperbolic system initially and also matches for the transport problem. By means of these subspaces, the Lax and Phillips semigroup is defined and it is proved that this semigroup is eventually compact, hence hyperbolic. 

Balanced equations give rise to two transport equations, one of which can satisfy an advection equation and one of which will be nonautonomous. For generating, the Howland semigroup and Howland’s formalism must be used, as shown in Chapter 5. 

Chapter 6 is the highlight of the book, in which it is explained how the scattering operator for the transport problem by using the albedo operator can lead to recovery of the functionality of computerized tomography in medical science.  The final chapter introduces the Wigner function, which connects the Schrödinger equation to statistical physics and the Husimi distribution function. Here, the relationship between the Wigner function and the quantum dynamical semigroup (QDS) can be seen.

✦ Table of Contents

Foreword by Peter Lax
Preface
Introduction
Contents
1 Semigroups of Linear Operators
1.1 Closed Operators
1.2 Strongly Continuous Semigroups
1.3 Dissipative Operators
1.4 Abstract Cauchy Problem
1.5 Hille–Yosida Theorem
1.6 The Spectral Properties of the Generator of a (C0) Semigroup
1.7 (C0) Semigroups in a Hilbert Space
1.8 Perturbation Theory
1.9 Positive Semigroups
1.10 Compact Semigroups
1.11 Hyperbolic and Chaotic Semigroups
1.12 Family of Strongly Continuous Propagators
1.13 The Howland Semigroup
1.14 Schatten–von Neumann Classes
1.15 Tensor Product Semigroup
1.16 Some Facts on Ultraweak Operator Topology
1.17 Quantum Dynamical Semigroup
1.18 Notes and Comments
2 Wave and Scattering Operators
2.1 Basic Properties of the Wave and Scattering Operators
2.2 Theorems of Existence for the Wave Operators
2.3 Smooth Perturbations
2.4 Contribution of Positivity
2.5 Limiting Absorption Principle
2.6 What Can We Obtain from the Existence of the Wave Operators?
2.7 Notes and Comments
3 Existence of the Wave Operators for the Transport Equation
3.1 Statement of the Time-Dependent Problem
3.2 Streaming Free Group
3.3 Absorption Group
3.4 Transport Group
3.5 Nonproliferating Systems
3.6 Existence of the Scattering Operator
3.7 Existence of the Other Wave Operators
3.8 Locally Decaying Property
3.9 Finite Collision Systems
3.10 Notes and Comments
4 The Lax and Phillips Formalism for the Transport Problems
4.1 The Lax–Phillips Representation Theorem
4.2 The Lax–Phillips Semigroups
4.3 Generalized Eigenfunction Expansion
4.4 Hypercyclicity in the Lax–Phillips Formalism
4.5 Notes and Comments
5 Scattering Theory for a Charged Particle Transport Problem
5.1 Derivation of the Charged Particle Transport Equation
5.2 Strongly Continuous Propagator for Charged Particle Transport Equation
5.3 Wave Operators for Charged Particle Transport Equation
5.4 Characterization of the Evolution Group
5.5 Howland's Formalism
5.6 Similarity Between Perturbed and Unperturbed Operators
5.7 Notes and Comments
6 Relationship Between the Albedo and Scattering Operators
6.1 Scattering Theory in Computerized Tomography
6.2 The Trace Theorems
6.3 Relationship Between the Albedo and Scattering Operators for Transparent Boundary Condition
6.4 Relationship Between the Albedo and Scattering Operators for Semi-transparent Boundary Condition
6.5 Notes and Comments
7 Scattering Theory for Quantum Transport Equation
7.1 Wigner Transform: A Bridge Between Quantum and Statistical Mechanics
7.2 Well-Posedness of the Wigner Problem
7.3 Liouville–Vlasov Equation
7.4 Wigner Function in the Weyl Calculus
7.5 Semiclassical Limit of Wigner Equation
7.6 Husimi Representation and Well-Posedness
7.7 Quantum Transport Equation
7.8 Scattering Theory for Quantum Liouville Equation
7.9 Scattering Theory for Quantum Dynamical Semigroup
7.10 Notes and Comments
Appendix List of Symbols
Bibliography
Index