**Graduate Studies in Mathematics**

Volume: 200;
2019;
634 pp;
Hardcover

MSC: Primary 58; 35; 34; 81;

**Print ISBN: 978-1-4704-4366-5
Product Code: GSM/200**

List Price: $95.00

AMS Member Price: $76.00

MAA Member Price: $85.50

**Electronic ISBN: 978-1-4704-5313-8
Product Code: GSM/200.E**

List Price: $95.00

AMS Member Price: $76.00

MAA Member Price: $85.50

#### Supplemental Materials

# Mathematical Theory of Scattering Resonances

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*Semyon Dyatlov; Maciej Zworski*

Scattering resonances generalize bound states/eigenvalues for
systems in which energy can scatter to infinity. A typical resonance
has a rate of oscillation (just as a bound state does) and a rate of
decay. Although the notion is intrinsically dynamical, an elegant
mathematical formulation comes from considering meromorphic
continuations of Green's functions. The poles of these meromorphic
continuations capture physical information by identifying the rate of
oscillation with the real part of a pole and the rate of decay with
its imaginary part. An example from mathematics is given by the zeros
of the Riemann zeta function: they are, essentially, the resonances of
the Laplacian on the modular surface. The Riemann hypothesis then
states that the decay rates for the modular surface are all either \(0\)
or \(\frac14\). An example from physics is given by quasi-normal modes
of black holes which appear in long-time asymptotics of gravitational
waves.

This book concentrates mostly on the simplest case of scattering by
compactly supported potentials but provides pointers to modern
literature where more general cases are studied. It also presents a
recent approach to the study of resonances on asymptotically
hyperbolic manifolds. The last two chapters are devoted to
semiclassical methods in the study of resonances.

#### Readership

Graduate students and researchers interested in scattering resonances.

#### Reviews & Endorsements

This is an up to date account of modern mathematical scattering theory with an emphasis on the deep interplay between the location of the scattering poles or resonances, and the underlying dynamics and geometry. The masterful exposition reflects the authors' significant roles in shaping this very active field. A must read for researchers and students working in scattering theory or related areas.

-- Peter Sarnak, Institute for Advanced Study

This is a very broad treatise of the modern theory of scattering resonances, beautifully written with a wealth of important mathematical results as well as applications, motivations and numerical and experimental illustrations. For experts, it will be a basic reference and for non-experts and graduate students an appealing and quite accessible introduction to a fascinating field with multiple connections to other branches of mathematics and to physics.

-- Johannes Sjöstrand, Université de Bourgogne

Resonance is the Queen of the realm of waves. No other book addresses this realm so completely and compellingly, oscillating effortlessly between illustration, example, and rigorous mathematical discourse. Mathematicians will find a wonderful array of physical phenomena given a solid intuitive and mathematical foundation, linked to deep theorems. Physicists and engineers will be inspired to consider new realms and phenomena. Chapters travel between motivation, light mathematics, and deeper mathematics, passing the baton from one to the other and back in a way that these authors are uniquely qualified to do.

-- Eric J. Heller, Harvard University

#### Table of Contents

# Table of Contents

## Mathematical Theory of Scattering Resonances

- Cover Cover11
- Title page iii4
- Preface ix10
- Chapter 1. Introduction 114
- Part 1 . POTENTIAL SCATTERING 1932
- Chapter 2. Scattering resonances in dimension one 2134
- 2.1. Outgoing and incoming solutions 2235
- 2.2. Meromorphic continuation 2639
- 2.3. Expansions of scattered waves 3952
- 2.4. Scattering matrix in dimension one 4558
- 2.5. Asymptotics for the counting function 5265
- 2.6. Trace and Breit–Wigner formulas 5972
- 2.7. Complex scaling in one dimension 7083
- 2.8. Semiclassical study of resonances 8295
- 2.9. Notes 91104
- 2.10. Exercises 92105

- Chapter 3. Scattering resonances in odd dimensions 95108
- 3.1. Free resolvent in odd dimensions 96109
- 3.2. Meromorphic continuation 108121
- 3.3. Resolvent at zero energy 116129
- 3.4. Upper bounds on the number of resonances 125138
- 3.5. Complex-valued potentials with no resonances 129142
- 3.6. Outgoing solutions and Rellich’s theorem 131144
- 3.7. The scattering matrix 143156
- 3.8. More on distorted plane waves 155168
- 3.9. The Birman–Kreĭn trace formula 159172
- 3.10. The Melrose trace formula 177190
- 3.11. Scattering asymptotics 187200
- 3.12. Existence of resonances for real potentials 205218
- 3.13. Notes 207220
- 3.14. Exercises 210223

- Part 2 . GEOMETRIC SCATTERING 215228
- Chapter 4. Black box scattering in \RR^{𝑛} 217230
- Chapter 5. Scattering on hyperbolic manifolds 305318
- 5.1. Asymptotically hyperbolic manifolds 307320
- 5.2. A motivating example 314327
- 5.3. The modified Laplacian 317330
- 5.4. Phase space dynamics 323336
- 5.5. Propagation estimates 332345
- 5.6. Meromorphic continuation 341354
- 5.7. Applications to general relativity 351364
- 5.8. Notes 362375
- 5.9. Exercises 364377

- Part 3 . RESONANCES IN THE SEMICLASSICAL LIMIT 369382
- Part 4 . APPENDICES 473486
- Back Cover Back Cover1649