Dispersion Decay and Scattering Theory
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- Published: 31 July 2012
- ISBN: 9781118341827
- Author(s): Alexander Komech, Elena Kopylova
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A simplified, yet rigorous treatment of scattering theory methods and their applications
Dispersion Decay and Scattering Theory provides thorough, easy-to-understand guidance on the application of scattering theory methods to modern problems in mathematics, quantum physics, and mathematical physics. Introducing spectral methods with applications to dispersion time-decay and scattering theory, this book presents, for the first time, the Agmon-Jensen-Kato spectral theory for the Schr?dinger equation, extending the theory to the Klein-Gordon equation. The dispersion decay plays a crucial role in the modern application to asymptotic stability of solitons of nonlinear Schr?dinger and Klein-Gordon equations.
The authors clearly explain the fundamental concepts and formulas of the Schr?dinger operators, discuss the basic properties of the Schr?dinger equation, and offer in-depth coverage of Agmon-Jensen-Kato theory of the dispersion decay in the weighted Sobolev norms. The book also details the application of dispersion decay to scattering and spectral theories, the scattering cross section, and the weighted energy decay for 3D Klein-Gordon and wave equations. Complete streamlined proofs for key areas of the Agmon-Jensen-Kato approach, such as the high-energy decay of the resolvent and the limiting absorption principle are also included.
Dispersion Decay and Scattering Theory is a suitable book for courses on scattering theory, partial differential equations, and functional analysis at the graduate level. The book also serves as an excellent resource for researchers, professionals, and academics in the fields of mathematics, mathematical physics, and quantum physics who would like to better understand scattering theory and partial differential equations and gain problem-solving skills in diverse areas, from high-energy physics to wave propagation and hydrodynamics. List of Figures xiii
Foreword xv
Preface xvii
Acknowledgments xix
Introduction xxi
1 Basic Concepts and Formulas 1
1 Distributions and Fourier transform 1
2 Functional spaces 3
2.1 Sobolev spaces 3
2.2 AgmonSobolev weighted spaces 4
2.3 Operatorvalued functions 5
3 Free propagator 6
3.1 Fourier transform 6
3.2 Gaussian integrals 8
2 Nonstationary Schrödinger Equation 11
4 Definition of solution 11
5 Schrödinger operator 14
5.1 A priori estimate 14
5.2 Hermitian symmetry 14
6 Dynamics for free Schrödinger equation 15
7 Perturbed Schrödinger equation 17
7.1 Reduction to integral equation 17
7.2 Contraction mapping 19
7.3 Unitarity and energy conservation 20
8 Wave and scattering operators 22
8.1 Möller wave operators. Cook method 22
8.2 Scattering operator 23
8.3 Intertwining identities 24
3 Stationary Schrödinger Equation 25
9 Free resolvent 25
9.1 General properties 25
9.2 Integral representation 28
10 Perturbed resolvent 31
10.1 Reduction to compact perturbation 31
10.2 Fredholm Theorem 32
10.3 Perturbation arguments 33
10.4 Continuous spectrum 35
10.5 Some improvements 36
4 Spectral Theory 37
11 Spectral representation 37
11.1 Inversion of Fourier-Laplace transform 37
11.2 Stationary Schrödinger equation 39
11.3 Spectral representation 39
11.4 Commutation relation 40
12 Analyticity of resolvent 41
13 Gohberg-Bleher theorem 43
14 Meromorphic continuation of resolvent 47
15 Absence of positive eigenvalues 50
15.1 Decay of eigenfunctions 50
15.2 Carleman estimates 54
15.3 Proof of Kato Theorem 56
5 High Energy Decay of Resolvent 59
16 High energy decay of free resolvent 59
16.1 Resolvent estimates 60
16.2 Decay of free resolvent 64
16.3 Decay of derivatives 65
17 High energy decay of perturbed resolvent 67
6 Limiting Absorption Principle 71
18 Free resolvent 71
19 Perturbed resolvent 77
19.1 The case λ > 0 77
19.2 The case λ = 0 78
20 Decay of eigenfunctions 81
20.1 Zero trace 81
20.2 Division problem 83
20.3 Negative eigenvalues 86
20.4 Appendix A: Sobolev Trace Theorem 86
20.5 Appendix B: SokhotskyPlemelj formula 87
7 Dispersion Decay 89
21 Proof of dispersion decay 90
22 Low energy asymptotics 92
8 Scattering Theory and Spectral Resolution 97
23 Scattering theory 97
23.1 Asymptotic completeness 97
23.2 Wave and scattering operators 99
23.3 Intertwining and commutation relations 99
24 Spectral resolution 101
24.1 Spectral resolution for the Schrödinger operator 101
24.2 Diagonalization of scattering operator 101
25 T Operator and SMatrix 1003
9 Scattering Cross Section 111
26 Introduction 111
27 Main results 117
28 Limiting Amplitude Principle 120
29 Spherical waves 121
30 Plane wave limit 125
31 Convergence of flux 127
32 Long range asymptotics 128
33 Cross section 131
10 Klein-Gordon Equation 133
35 Introduction 134
36 Free Klein-Gordon equation 137
36.1 Dispersion decay 137
36.2 Spectral properties 139
37 Perturbed Klein-Gordon equation 143
37.1 Spectral properties 143
37.2 Dispersion decay 145
38 Asymptotic completeness 149
11 Wave equation 151
39 Introduction 152
40 Free wave equation 154
40.1 Time-decay 154
40.2 Spectral properties 155
41 Perturbed wave equation 158
41.1 Spectral properties 158
41.2 Dispersion decay 160
42 Asymptotic completeness 163
43 Appendix: Sobolev embedding theorem 165
References 167
Index 172
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