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E-raamat: Mathematical Theory of Scattering Resonances

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Mathematical Theory of Scattering ResonancesSuurem pilt
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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.

Arvustused

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 Sjostrand, Universite 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

Preface ix
Chapter 1 Introduction
1(20)
§1.1 Resonances in scattering theory
1(6)
§1.2 Semiclassical study of resonances
7(1)
§1.3 Some examples
8(5)
§1.4 Overview
13(8)
Part 1 POTENTIAL SCATTERING
Chapter 2 Scattering resonances in dimension one
21(74)
§2.1 Outgoing and incoming solutions
22(4)
§2.2 Meromorphic continuation
26(13)
§2.3 Expansions of scattered waves
39(6)
§2.4 Scattering matrix in dimension one
45(7)
§2.5 Asymptotics for the counting function
52(7)
§2.6 Trace and Breit--Wigner formulas
59(11)
§2.7 Complex scaling in one dimension
70(12)
§2.8 Semiclassical study of resonances
82(9)
§2.9 Notes
91(1)
§2.10 Exercises
92(3)
Chapter 3 Scattering resonances in odd dimensions
95(122)
§3.1 Free resolvent in odd dimensions
96(12)
§3.2 Meromorphic continuation
108(8)
§3.3 Resolvent at zero energy
116(9)
§3.4 Upper bounds on the number of resonances
125(4)
§3.5 Complex-valued potentials with no resonances
129(2)
§3.6 Outgoing solutions and Rellich's theorem
131(12)
§3.7 The scattering matrix
143(12)
§3.8 More on distorted plane waves
155(4)
§3.9 The Birman--Krein trace formula
159(18)
§3.10 The Melrose trace formula
177(10)
§3.11 Scattering asymptotics
187(18)
§3.12 Existence of resonances for real potentials
205(2)
§3.13 Notes
207(3)
§3.14 Exercises
210(7)
Part 2 GEOMETRIC SCATTERING
Chapter 4 Black box scattering in Rn
217(88)
§4.1 General assumptions
218(5)
§4.2 Meromorphic continuation
223(12)
§4.3 Upper bounds on the number of resonances
235(15)
§4.4 Plane waves and the scattering matrix
250(18)
§4.5 Complex scaling
268(21)
§4.6 Singularities and resonance-free regions
289(11)
§4.7 Notes
300(3)
§4.8 Exercises
303(2)
Chapter 5 Scattering on hyperbolic manifolds
305(66)
§5.1 Asymptotically hyperbolic manifolds
307(7)
§5.2 A motivating example
314(3)
§5.3 The modified Laplacian
317(6)
§5.4 Phase space dynamics
323(9)
§5.5 Propagation estimates
332(9)
§5.6 Meromorphic continuation
341(10)
§5.7 Applications to general relativity
351(11)
§5.8 Notes
362(2)
§5.9 Exercises
364(7)
Part 3 RESONANCES IN THE SEMICLASSICAL LIMIT
Chapter 6 Resonance-free regions
371(54)
§6.1 Geometry of trapping
373(7)
§6.2 Resonances in strips
380(12)
§6.3 Normally hyperbolic trapping
392(11)
§6.4 Logarithmic resonance-free regions
403(5)
§6.5 Lower bounds on resonance widths
408(10)
§6.6 Notes
418(3)
§6.7 Exercises
421(4)
Chapter 7 Resonances and trapping
425(50)
§7.1 Lower bounds on the resolvent
426(6)
§7.2 Semiclassical growth estimates
432(5)
§7.3 From quasimodes to resonances
437(9)
§7.4 The Sjostrand trace formula
446(10)
§7.5 Resonance expansions for strong trapping
456(11)
§7.6 Notes
467(1)
§7.7 Exercises
468(7)
Part 4 APPENDICES
Appendix A Notation
475(8)
§A.1 Basic notation
475(1)
§A.2 Functions
476(1)
§A.3 Spaces of functions
477(1)
§A.4 Operators
477(1)
§A.5 Estimates
478(1)
§A.6 Tempered distributions
479(1)
§A.7 Distributions on manifolds and Schwartz kernels
480(3)
Appendix B Spectral theory
483(24)
§B.1 Spectral theory of self-adjoint operators
483(5)
§B.2 Functional calculus
488(1)
§B.3 Singular values
489(3)
§B.4 The trace class
492(5)
§B.5 Weyl inequalities and Fredholm determinants
497(7)
§B.6 Lidskii's theorem
504(2)
§B.7 Notes
506(1)
§B.8 Exercises
506(1)
Appendix C Fredholm theory
507(20)
§C.1 Grushin problems
507(2)
§C.2 Fredholm operators
509(4)
§C.3 Meromorphic continuation of operators
513(3)
§C.4 Gohberg--Sigal theory
516(8)
§C.5 Notes
524(1)
§C.6 Exercises
524(3)
Appendix D Complex analysis
527(8)
§D.1 General facts
527(4)
§D.2 Entire functions
531(4)
Appendix E Semiclassical analysis
535(78)
§E.1 Pseudodifferential operators
536(20)
§E.2 Wavefront sets and ellipticity
556(10)
§E.3 Semiclassical defect measures
566(3)
§E.4 Propagation estimates
569(17)
§E.5 Hyperbolic estimates
586(17)
§E.6 Notes
603(1)
§E.7 Exercises
604(9)
Bibliography 613(18)
Index 631
Semyon Dyatlov, University of California, Berkeley, CA, and MIT, Cambridge, MA.

Maciej Zworski, University of California, Berkeley, CA.
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