Dispersion Decay and Scattering Theory.
Title:
Dispersion Decay and Scattering Theory.
Author:
Komech, Alexander.
ISBN:
9781118382899
Personal Author:
Edition:
1st ed.
Physical Description:
1 online resource (204 pages)
Contents:
Dispersion Decay and Scattering Theory -- CONTENTS -- List of Figures -- Foreword -- Preface -- Acknowledgments -- Introduction -- 1 Basic Concepts and Formulas -- 1 Distributions and Fourier transform -- 2 Functional spaces -- 2.1 Sobolev spaces -- 2.2 Agmon-Sobolev weighted spaces -- 2.3 Operator-valued functions -- 3 Free propagator -- 3.1 Fourier transform -- 3.2 Gaussian integrals -- 2 Nonstationary Schrodinger Equation -- 4 Definition of solution -- 5 Schrödinger operator -- 5.1 A priori estimate -- 5.2 Hermitian symmetry -- 6 Dynamics for free Schrödinger equation -- 7 Perturbed Schrödinger equation -- 7.1 Reduction to integral equation -- 7.2 Contraction mapping -- 7.3 Unitarity and energy conservation -- 8 Wave and scattering operators -- 8.1 Möller wave operators: Cook method -- 8.2 Scattering operator -- 8.3 Intertwining identities -- 3 Stationary Schrödinger Equation -- 9 Free resolvent -- 9.1 General properties -- 9.2 Integral representation -- 10 Perturbed resolvent -- 10.1 Reduction to compact perturbation -- 10.2 Fredholm Theorem -- 10.3 Perturbation arguments -- 10.4 Continuous spectrum -- 10.5 Some improvements -- 4 Spectral Theory -- 11 Spectral representation -- 11.1 Inversion of Fourier-Laplace transform -- 11.2 Stationary Schrödinger equation -- 11.3 Spectral representation -- 11.4 Commutation relation -- 12 Analyticity of resolvent -- 13 Gohberg-Bleher theorem -- 14 Meromorphic continuation of resolvent -- 15 Absence of positive eigenvalues -- 15.1 Decay of eigenfunctions -- 15.2 Carleman estimates -- 15.3 Proof of Kato Theorem -- 5 High Energy Decay of Resolvent -- 16 High energy decay of free resolvent -- 16.1 Resolvent estimates -- 16.2 Decay of free resolvent -- 16.3 Decay of derivatives -- 17 High energy decay of perturbed resolvent -- 6 Limiting Absorption Principle -- 18 Free resolvent -- 19 Perturbed resolvent.
19.1 The case λ > 0 -- 19.2 The case λ = 0 -- 20 Decay of eigenfunctions -- 20.1 Zero trace -- 20.2 Division problem -- 20.3 Negative eigenvalues -- 20.4 Appendix A: Sobolev Trace Theorem -- 20.5 Appendix B: Sokhotsky-Plemelj formula -- 7 Dispersion Decay -- 21 Proof of dispersion decay -- 22 Low energy asymptotics -- 8 Scattering Theory and Spectral Resolution -- 23 Scattering theory -- 23.1 Asymptotic completeness -- 23.2 Wave and scattering operators -- 23.3 Intertwining and commutation relations -- 24 Spectral resolution -- 24.1 Spectral resolution for the Schrödinger operator -- 24.2 Diagonalization of scattering operator -- 25 T-Operator and 5-Matrix -- 9 Scattering Cross Section -- 26 Introduction -- 27 Main results -- 28 Limiting amplitude principle -- 29 Spherical waves -- 30 Plane wave limit -- 31 Convergence of flux -- 32 Long range asymptotics -- 33 Cross section -- 10 Klein-Gordon Equation -- 34 Introduction -- 35 Free Klein-Gordon equation -- 35.1 Dispersion decay -- 35.2 Spectral properties -- 36 Perturbed Klein-Gordon equation -- 36.1 Spectral properties -- 36.2 Dispersion decay -- 37 Asymptotic completeness -- 11 Wave equation -- 38 Introduction -- 39 Free wave equation -- 39.1 Time decay -- 39.2 Spectral properties -- 40 Perturbed wave equation -- 40.1 Spectral properties -- 40.2 Dispersion decay -- 41 Asymptotic completeness -- 42 Appendix: Sobolev Embedding Theorem -- References -- Index.
Abstract:
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.
Local Note:
Electronic reproduction. Ann Arbor, Michigan : ProQuest Ebook Central, 2017. Available via World Wide Web. Access may be limited to ProQuest Ebook Central affiliated libraries.
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