**Mathematical Surveys and Monographs**

Volume: 28;
1988;
209 pp;
Softcover

MSC: Primary 58;
Secondary 34; 35; 81

Print ISBN: 978-1-4704-2054-3

Product Code: SURV/28.S

List Price: $84.00

Individual Member Price: $67.20

**Electronic ISBN: 978-1-4704-1255-5
Product Code: SURV/28.E**

List Price: $84.00

Individual Member Price: $67.20

# Direct and Inverse Scattering on the Line

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*Richard Beals; Percy Deift; Carlos Tomei*

This book deals with the theory of linear ordinary differential operators of arbitrary order. Unlike treatments that focus on spectral theory, this work centers on the construction of special eigenfunctions (generalized Jost solutions) and on the inverse problem: the problem of reconstructing the operator from minimal data associated to the special eigenfunctions. In the second order case this program includes spectral theory and is equivalent to quantum mechanical scattering theory; the essential analysis involves only the bounded eigenfunctions. For higher order operators, bounded eigenfunctions are again sufficient for spectral theory and quantum scattering theory, but they are far from sufficient for a successful inverse theory.

The authors give a complete and self-contained theory of the inverse problem for an ordinary differential operator of any order. The theory provides a linearization for the associated nonlinear evolution equations, including KdV and Boussinesq. The authors also discuss Darboux-Bäcklund transformations, related first-order systems and their evolutions, and applications to spectral theory and quantum mechanical scattering theory.

Among the book's most significant contributions are a new construction of normalized eigenfunctions and the first complete treatment of the self-adjoint inverse problem in order greater than two. In addition, the authors present the first analytic treatment of the corresponding flows, including a detailed description of the phase space for Boussinesq and other equations.

The book is intended for mathematicians, physicists, and engineers in the area of soliton equations, as well as those interested in the analytical aspects of inverse scattering or in the general theory of linear ordinary differential operators. This book is likely to be a valuable resource to many.

Required background consists of a basic knowledge of complex variable theory, the theory of ordinary differential equations, linear algebra, and functional analysis. The authors have attempted to make the book sufficiently complete and self-contained to make it accessible to a graduate student having no prior knowledge of scattering or inverse scattering theory. The book may therefore be suitable for a graduate textbook or as background reading in a seminar.

#### Table of Contents

# Table of Contents

## Direct and Inverse Scattering on the Line

- Cover Cover11 free
- Title page iii4 free
- Contents vii8 free
- Preface xiii14 free
- Introduction 116 free
- Part I. The Forward Problem 520 free
- 1. Distinguished Solutions 722
- 2. Fundamental Matrices 924
- 2 bis. The Second Order Case 1328
- 3. Fundamental Tensors 1732
- 4. Behavior of Fundamental Tensors as |𝑥|→∞; the Functions 𝛿_{𝑘} 2136
- 5. Behavior of Fundamental Tensors as 𝑧→∞ 2237
- 6. Behavior of Fundamental Tensors as 𝑧→0 2439
- 7. Construction of Fundamental Matrices 2843
- 8. Global Properties of Fundamental Matrices; the Transition Matrix 𝛿 3348
- 9. Symmetries of Fundamental Matrices 4055
- 10. The Green’s Function for 𝐿 4257
- 11. Generic Operators and Scattering Data 4459
- 12. Algebraic Properties of Scattering Data 4661
- 13. Analytic Properties of Scattering Data 5267
- 14. Scattering Data for 𝑚; Determination of 𝑣 from 𝑣 5469
- 15. Scattering Data for 𝐿* 5772
- 16. Generic Selfadjoint Operators and Scattering Data 5873
- 17. The Green’s Function Revisited 6580
- 18. Genericity at 𝑧=0 6782
- 19. Genericity at 𝑧≠0 7085
- 20. Summary of Properties of Scattering Data 7792
- Part II. The Inverse Problem 8196
- 21. Normalized Eigenfunctions for Odd Order Inverse Data 8499
- 22. The Vanishing Lemma 86101
- 23. The Cauchy Operator 87102
- 24. Equations for the Inverse Problem 91106
- 25. Factorization of 𝑣 near 𝑧=0 and Property (20.6) 98113
- 26. Reduction to a Fredholm Equation 104119
- 27. Existence of ℎ^{#} 114129
- 28. Properties of ℎ^{#} 117132
- 29. Properties of 𝜇^{#}(𝑥,𝑧) and 𝜇(𝑥,𝑧) as 𝑧→∞ and as 𝑥→-∞ 123138
- 30. Proof of the Basic Inverse Theorem 127142
- 31. The Scalar Factorization Problem for 𝛿 130145
- 32. The Inverse Problem at 𝑥=+∞ and the bijectivity of the map 𝐿\mapsto𝑆(𝐿)=(𝑍(𝐿),𝑣(𝐿)) 134149
- 33. The Even Order Case 137152
- 34. The Second Order Problem 143158
- Part III. Applications 147162
- 35. Flows 149164
- 36. Eigenfunction Expansions and Classical Scattering Theory 162177
- 37. Inserting and Removing Poles 170185
- 38. Matrix Factorization and First Order Systems 181196
- Appendix A. Rational approximation 197212
- Appendix B. Some formulas 201216
- References 203218
- Notation Index 207222
- Index 209224 free
- Back Cover Back Cover1226