**Mathematical Surveys and Monographs**

Volume: 158;
2010;
444 pp;
Hardcover

MSC: Primary 34; 35; 47; 81;

Print ISBN: 978-0-8218-0331-8

Product Code: SURV/158

List Price: $110.00

Individual Member Price: $88.00

**Electronic ISBN: 978-1-4704-1385-9
Product Code: SURV/158.E**

List Price: $110.00

Individual Member Price: $88.00

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#### Supplemental Materials

# Mathematical Scattering Theory: Analytic Theory

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*D. R. Yafaev*

The main subject of this book is applications of methods of scattering theory to differential operators, primarily the Schrödinger operator.

There are two different trends in scattering theory for
differential operators. The first one relies on the abstract
scattering theory. The second one is almost independent of it. In this
approach the abstract theory is replaced by a concrete investigation
of the corresponding differential equation. In this book both of these
trends are presented. The first half of this book begins with the
summary of the main results of the general scattering theory of the
previous book by the author,
*Mathematical Scattering Theory: General Theory*, American
Mathematical Society, 1992. The next three chapters illustrate basic theorems
of abstract scattering theory, presenting, in particular, their applications to
scattering theory of perturbations of differential operators with constant
coefficients and to the analysis of the trace class method.

In the second half of the book direct methods of scattering theory for differential operators are presented. After considering the one-dimensional case, the author returns to the multi-dimensional problem and discusses various analytical methods and tools appropriate for the analysis of differential operators, including, among others, high- and low-energy asymptotics of the Green function, the scattering matrix, ray and eikonal expansions.

The book is based on graduate courses taught by the author at Saint-Petersburg (Russia) and Rennes (France) Universities and is oriented towards a reader interested in studying deep aspects of scattering theory (for example, a graduate student in mathematical physics).

#### Table of Contents

# Table of Contents

## Mathematical Scattering Theory: Analytic Theory

- Contents vii8 free
- Preface xi12 free
- Basic Notation 116 free
- Introduction 520
- Chapter 0. Basic Concepts 1732
- Chapter 1. Smooth Theory. The Schrodinger Operator 7186
- Chapter 2. Smooth Theory. General Differential Operators 109124
- Chapter 3. Scattering for Perturbations of Trace Class Type 133148
- Chapter 4. Scattering on the Half-line 161176
- Chapter 5. One-Dimensional Scattering 209224
- Chapter 6. The Limiting Absorption Principle (LAP)... 231246
- Chapter 7. High- and Low-Energy Asymptotics 267282
- Chapter 8. The Scattering Matrix (SM) and the Scattering Cross Section 297312
- Chapter 9. The Spectral Shift Function and the Trace Formulas 341356
- Chapter 10. The Schrodinger Operator with a Long-Range Potential 369384
- Chapter 11. The LAP and Radiation Estimates Revisited 399414
- Review of the Literature 415430
- Bibliography 429444
- Index 441456 free

#### Readership

Graduate students and research mathematicians interested in spectral theory of differential operators.