Direct and Inverse Scattering for the Matrix Schrödinger Equation (Applied Mathematical Sciences, 203)

Preface
Acknowledgments
Contents
1 Introduction
	1.1 An Overview
	1.2 Notes on the Bibliography
2 The Matrix Schrödinger Equation and the Characterization of the Scattering Data
	2.1 Outline of the Chapter
	2.2 The Matrix Schrödinger Equation on the Half Line
	2.3 Star Graphs
	2.4 The Schrödinger Equation on the Full Line
	2.5 The Faddeev Class and the Marchenko Class
	2.6 A First Characterization of the Scattering Data
	2.7 Alternate Characterizations of the Scattering Data
	2.8 Another Characterization of the Scattering Data
3 Direct Scattering I
	3.1 Outline of the Solution to the Direct Problem
	3.2 Special Solutions to the Schrödinger Equation
	3.3 The Hamiltonian
	3.4 Equivalence of the Formulations of the Boundary Condition
	3.5 The Quadratic Form of the Hamiltonian
	3.6 Transformations of the Jost and Scattering Matrices
	3.7 The Jost and Scattering Matrices with Zero Potential
	3.8 Low-Energy Analysis with Potentials in the Faddeev Class
	3.9 Low-Energy Analysis with Potentials of FiniteSecond Moment
	3.10 High-Energy Analysis
	3.11 Bound States
	3.12 Levinson\'s Theorem
	3.13 Further Properties of the Scattering Data
	3.14 The Marchenko Integral Equation
	3.15 The Boundary Matrices
	3.16 The Existence and Uniqueness in the Direct Problem
4 Direct Scattering II
	4.1 Basic Principles of the Scattering Theory
	4.2 The Limiting Absorption Principle
	4.3 The Generalized Fourier Maps for the Absolutely Continuous Subspace
	4.4 The Wave Operators
	4.5 The Scattering Operator and the Scattering Matrix
	4.6 The Spectral Shift Function
	4.7 Trace Formulas
	4.8 The Number of Bound States
5 Inverse Scattering
	5.1 Nonuniqueness Due to the Improperly Defined ScatteringMatrix
	5.2 The Solution to the Inverse Problem
	5.3 Bounds on the Constructed Solutions
	5.4 Relations Among the Characterization Conditions
	5.5 The Proof of the First Characterization Theorem
	5.6 Equivalents for Some Characterization Conditions
	5.7 Inverse Problem Using Only the Scattering Matrix as Input
	5.8 Characterization via Levinson\'s Theorem
	5.9 Parseval\'s Equality
	5.10 The Generalized Fourier Map
	5.11 An Alternate Method to Solve the Inverse Problem
	5.12 Characterization with Potentials of Stronger Decay
	5.13 The Dirichlet Boundary Condition
6 Some Explicit Examples
	6.1 Illustration of the Theory with Explicit Examples
	6.2 Some Methods Yielding Explicit Examples
	6.3 Explicit Examples in the Characterization of the Scattering Data
	6.4 Explicit Examples of Particular Solutions
Appendix A Mathematical Preliminaries
	A.1 Vectors, Matrices, and Functions
	A.2 Banach and Hilbert Spaces
	A.3 Inequalities
	A.4 Mollifiers
	A.5 Equicontinuity
	A.6 Distributions
	A.7 Absolute Continuity
	A.8 Sobolev Spaces
	A.9 The Fourier Transform
	A.10 Hardy Spaces
	A.11 Other Banach Spaces
	A.12 Linear Operators Between Banach and Hilbert Spaces
	A.13 Operators Between Finite Dimensional Hilbert Spaces
	A.14 Self-adjoint Operators and Symmetric Quadratic Forms
	A.15 Trace-Class and Hilbert–Schmidt Operators
	A.16 Resolvent and Spectrum
	A.17 The Spectral Theorem
	A.18 The Spectral Shift Function
	A.19 Deficiency Indices
	A.20 Self-adjoint Extensions of Matrix Schrödinger Operators
	A.21 Hermitian Symplectic Geometry
	A.22 Integral Operators
	A.23 The Gronwall Inequalities
	A.24 Miscellaneous Results
	A.25 Notes
References
Index
List of Symbols