Mathematical Theory of Scattering Resonances

Semyon Dyatlov, Maciej Zworski

• Publisher: American Mathematical Society
• Year: 2019
• Print ISBN: 9781470443665
• eBook ISBN: 9781470453138
• Topic: Mathematics
• Subtopic: Applied Mathematics
• Index: Mathematics

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.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 StudyThis 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 BourgogneResonance 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

• Cover
• Title page
• Preface
• Chapter 1. Introduction
• Part 1 . POTENTIAL SCATTERING
• Part 2 . GEOMETRIC SCATTERING
• Part 3 . RESONANCES IN THE SEMICLASSICAL LIMIT
• Part 4 . APPENDICES
• Back Cover