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Informationen zum Autor 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. Klappentext A simplified, yet rigorous treatment of scattering theory methods and their applicationsDispersion 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. Zusammenfassung 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. Inhaltsverzeichnis 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
