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

✍ Scribed by Tuncay Aktosun, Ricardo Weder

• Publisher: Springer
• Year: 2020
• Tongue: English
• Leaves: 631
• Category: Library

✦ Synopsis

Authored by two experts in the field who have been long-time collaborators, this monograph treats the scattering and inverse scattering problems for the matrix Schrödinger equation on the half line with the general selfadjoint boundary condition. The existence, uniqueness, construction, and characterization aspects are treated with mathematical rigor, and physical insight is provided to make the material accessible to mathematicians, physicists, engineers, and applied scientists with an interest in scattering and inverse scattering. The material presented is expected to be useful to beginners as well as experts in the field. The subject matter covered is expected to be interesting to a wide range of researchers including those working in quantum graphs and scattering on graphs. The theory presented is illustrated with various explicit examples to improve the understanding of scattering and inverse scattering problems.
The monograph introduces a specific class of input data sets consisting of a potential and a boundary condition and a specific class of scattering data sets consisting of a scattering matrix and bound-state information. The important problem of the characterization is solved by establishing a one-to-one correspondence between the two aforementioned classes. The characterization result is formulated in various equivalent forms, providing insight and allowing a comparison of different techniques used to solve the inverse scattering problem. The past literature treated the type of boundary condition as a part of the scattering data used as input to recover the potential. This monograph provides a proper formulation of the inverse scattering problem where the type of boundary condition is no longer a part of the scattering data set, but rather both the potential and the type of boundary condition are recovered from the scattering data set.

✦ Table of Contents

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