Dispersion Decay and Scattering Theory

Thoroughly classroom tested, this book applies scattering theory methods to modern problems within a variety of areas in advanced mathematics, quantum physics, and mathematical physics. It features the application of functional analysis, complex analysis, and theory of distributions for a comprehensive treatment of problems in mathematical physics and covers the Agmon-Jensen-Kato theory, along with eigenvalues, Klein-Gordon, and wave equations. Filled with exercises, hints, and explanatory figures throughout, this book will prove invaluable for students and professionals looking to expand their knowledge of scattering theory and partial differential equations.

Alexander Komech, PhD, is Professor and Senior Scientist in the Department of Mathematics at Vienna University and the Institute for Information Transmission Problems at the Russian Academy of Sciences. He is the author of more than 100 published journal articles. Elena Kopylova, PhD, is Senior Scientist in the Department of Mathematics at Vienna University and the institute for Information Transmission Problems at the Russian Academy of Sciences. She is the author of approximately 50 published journal articles.

List of Figures	p. xi
Foreword	p. xiii
Preface	p. xv
Acknowledgments	p. xvii
Introduction	p. xix
Basic Concepts and Formulas	p. 1
Distributions and Fourier transform	p. 1
Functional spaces	p. 3
Sobolev spaces	p. 3
Agmon-Sobolev weighted spaces	p. 4
Operator-valued functions	p. 5
Free propagator	p. 6
Fourier transform	p. 6
Gaussian integrals	p. 8
Nonstationary Schröodinger Equation	p. 11
Definition of solution	p. 11
Schrödinger operator	p. 14
A priori estimate	p. 14
Hermitian symmetry	p. 14
Dynamics for free Schrödinger equation	p. 15
Perturbed Schrödinger equation	p. 17
Reduction to integral equation	p. 17
Contraction mapping	p. 19
Unitarity and energy conservation	p. 20
Wave and scattering operators	p. 22
Möller wave operators: Cook method	p. 22
Scattering operator	p. 23
Intertwining identities	p. 24
Stationary Schrödinger Equation	p. 25
Free resolvent	p. 25
General properties	p. 25
Integral representation	p. 28
Perturbed resolvent	p. 31
Reduction to compact perturbation	p. 31
Fredholm Theorem	p. 32
Perturbation arguments	p. 33
Continuous spectrum	p. 35
Some improvements	p. 36
Spectral Theory	p. 37
Spectral representation	p. 37
Inversion of Fourier-Laplace transform	p. 37
Stationary Schrödinger equation	p. 39
Spectral representation	p. 39
Commutation relation	p. 40
Analyticity of resolvent	p. 41
Gohberg-Bleher theorem	p. 43
Meromorphic continuation of resolvent	p. 47
Absence of positive eigenvalues	p. 50
Decay of eigenfunctions	p. 50
Carleman estimates	p. 54
Proof of Kato Theorem	p. 56
High Energy Decay of Resolvent	p. 59
High energy decay of free resolvent	p. 59
Resolvent estimates	p. 60
Decay of free resolvent	p. 54
Decay of derivatives	p. 65
High energy decay of perturbed resolvent	p. 67
Limiting Absorption Principle	p. 71
Free resolvent	p. 71
Perturbed resolvent	p. 77
The case ¿ > 0	p. 77
The case ¿ = 0	p. 78
Decay of eigenfunctions	p. 81
Zero trace	p. 81
Division problem	p. 83
Negative eigenvalues	p. 86
Appendix A: Sobolev Trace Theorem	p. 87
Appendix B: Sokhotsky-Plemelj formula	p. 87
Dispersion Decay	p. 89
Proof of dispersion decay	p. 90
Low energy asymptotics	p. 92
Scattering Theory and Spectral Resolution	p. 97
Scattering theory	p. 97
Asymptotic completeness	p. 97
Wave and scattering operators	p. 99
Intertwining and commutation relations	p. 99
Spectral resolution	p. 101
Spectral resolution for the Schrödinger operator	p. 101
Diagonalization of scattering operator	p. 101
T-Operator and S-Matrix	p. 103
Scattering Cross Section	p. 111
Introduction	p. 111
Main results	p. 117
Limiting amplitude principle	p. 120
Spherical waves	p. 121
Plane wave limit	p. 125
Convergence of flux	p. 127
Long range asymptotics	p. 128
Cross section	p. 131
Klein-Gordon Equation	p. 133
Introduction	p. 134
Free Klein-Gordon equation	p. 137
Dispersion decay	p. 137
Spectral properties	p. 139
Perturbed Klein-Gordon equation	p. 143
Spectral properties	p. 143
Dispersion decay	p. 145
Asymptotic completeness	p. 149
Wave equation	p. 151
Introduction	p. 152
Free wave equation	p. 154
Time decay	p. 154
Spectral properties	p. 155
Perturbed wave equation	p. 158
Spectral properties	p. 158
Dispersion decay	p. 160
Asymptotic completeness	p. 163
Appendix: Sobolev Embedding Theorem	p. 165
References	p. 167
Index	p. 173
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