{"product_id":"dispersion-decay-and-scattering-theory-isbn-9781118341827","title":"Dispersion Decay and Scattering Theory","description":"\u003cp\u003e\u003cb\u003eA simplified, yet rigorous treatment of scattering theory methods and their applications\u003c\/b\u003e\u003c\/p\u003e \u003cp\u003e\u003ci\u003eDispersion Decay and Scattering Theory\u003c\/i\u003e 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.\u003c\/p\u003e \u003cp\u003eThe 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.\u003c\/p\u003e \u003cp\u003e\u003ci\u003eDispersion Decay and Scattering Theory\u003c\/i\u003e 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.\u003c\/p\u003e  List of Figures xiii  \u003cp\u003eForeword xv\u003c\/p\u003e \u003cp\u003ePreface xvii\u003c\/p\u003e \u003cp\u003eAcknowledgments xix\u003c\/p\u003e \u003cp\u003eIntroduction xxi\u003c\/p\u003e \u003cp\u003e\u003cb\u003e1 Basic Concepts and Formulas 1\u003c\/b\u003e\u003c\/p\u003e \u003cp\u003e1 Distributions and Fourier transform 1\u003c\/p\u003e \u003cp\u003e2 Functional spaces 3\u003c\/p\u003e \u003cp\u003e2.1 Sobolev spaces 3\u003c\/p\u003e \u003cp\u003e2.2 AgmonSobolev weighted spaces 4\u003c\/p\u003e \u003cp\u003e2.3 Operatorvalued functions 5\u003c\/p\u003e \u003cp\u003e3 Free propagator 6\u003c\/p\u003e \u003cp\u003e3.1 Fourier transform 6\u003c\/p\u003e \u003cp\u003e3.2 Gaussian integrals 8\u003c\/p\u003e \u003cp\u003e\u003cb\u003e2 Nonstationary Schrödinger Equation 11\u003c\/b\u003e\u003c\/p\u003e \u003cp\u003e4 Definition of solution 11\u003c\/p\u003e \u003cp\u003e5 Schrödinger operator 14\u003c\/p\u003e \u003cp\u003e5.1 A priori estimate 14\u003c\/p\u003e \u003cp\u003e5.2 Hermitian symmetry 14\u003c\/p\u003e \u003cp\u003e6 Dynamics for free Schrödinger equation 15\u003c\/p\u003e \u003cp\u003e7 Perturbed Schrödinger equation 17\u003c\/p\u003e \u003cp\u003e7.1 Reduction to integral equation 17\u003c\/p\u003e \u003cp\u003e7.2 Contraction mapping 19\u003c\/p\u003e \u003cp\u003e7.3 Unitarity and energy conservation 20\u003c\/p\u003e \u003cp\u003e8 Wave and scattering operators 22\u003c\/p\u003e \u003cp\u003e8.1 Möller wave operators. Cook method 22\u003c\/p\u003e \u003cp\u003e8.2 Scattering operator 23\u003c\/p\u003e \u003cp\u003e8.3 Intertwining identities 24\u003c\/p\u003e \u003cp\u003e\u003cb\u003e3 Stationary Schrödinger Equation 25\u003c\/b\u003e\u003c\/p\u003e \u003cp\u003e9 Free resolvent 25\u003c\/p\u003e \u003cp\u003e9.1 General properties 25\u003c\/p\u003e \u003cp\u003e9.2 Integral representation 28\u003c\/p\u003e \u003cp\u003e10 Perturbed resolvent 31\u003c\/p\u003e \u003cp\u003e10.1 Reduction to compact perturbation 31\u003c\/p\u003e \u003cp\u003e10.2 Fredholm Theorem 32\u003c\/p\u003e \u003cp\u003e10.3 Perturbation arguments 33\u003c\/p\u003e \u003cp\u003e10.4 Continuous spectrum 35\u003c\/p\u003e \u003cp\u003e10.5 Some improvements 36\u003c\/p\u003e \u003cp\u003e\u003cb\u003e4 Spectral Theory 37\u003c\/b\u003e\u003c\/p\u003e \u003cp\u003e11 Spectral representation 37\u003c\/p\u003e \u003cp\u003e11.1 Inversion of Fourier-Laplace transform 37\u003c\/p\u003e \u003cp\u003e11.2 Stationary Schrödinger equation 39\u003c\/p\u003e \u003cp\u003e11.3 Spectral representation 39\u003c\/p\u003e \u003cp\u003e11.4 Commutation relation 40\u003c\/p\u003e \u003cp\u003e12 Analyticity of resolvent 41\u003c\/p\u003e \u003cp\u003e13 Gohberg-Bleher theorem 43\u003c\/p\u003e \u003cp\u003e14 Meromorphic continuation of resolvent 47\u003c\/p\u003e \u003cp\u003e15 Absence of positive eigenvalues 50\u003c\/p\u003e \u003cp\u003e15.1 Decay of eigenfunctions 50\u003c\/p\u003e \u003cp\u003e15.2 Carleman estimates 54\u003c\/p\u003e \u003cp\u003e15.3 Proof of Kato Theorem 56\u003c\/p\u003e \u003cp\u003e\u003cb\u003e5 High Energy Decay of Resolvent 59\u003c\/b\u003e\u003c\/p\u003e \u003cp\u003e16 High energy decay of free resolvent 59\u003c\/p\u003e \u003cp\u003e16.1 Resolvent estimates 60\u003c\/p\u003e \u003cp\u003e16.2 Decay of free resolvent 64\u003c\/p\u003e \u003cp\u003e16.3 Decay of derivatives 65\u003c\/p\u003e \u003cp\u003e17 High energy decay of perturbed resolvent 67\u003c\/p\u003e \u003cp\u003e\u003cb\u003e6 Limiting Absorption Principle 71\u003c\/b\u003e\u003c\/p\u003e \u003cp\u003e18 Free resolvent 71\u003c\/p\u003e \u003cp\u003e19 Perturbed resolvent 77\u003c\/p\u003e \u003cp\u003e19.1 The case λ \u0026gt; 0 77\u003c\/p\u003e \u003cp\u003e19.2 The case λ = 0 78\u003c\/p\u003e \u003cp\u003e20 Decay of eigenfunctions 81\u003c\/p\u003e \u003cp\u003e20.1 Zero trace 81\u003c\/p\u003e \u003cp\u003e20.2 Division problem 83\u003c\/p\u003e \u003cp\u003e20.3 Negative eigenvalues 86\u003c\/p\u003e \u003cp\u003e20.4 Appendix A: Sobolev Trace Theorem 86\u003c\/p\u003e \u003cp\u003e20.5 Appendix B: SokhotskyPlemelj formula 87\u003c\/p\u003e \u003cp\u003e\u003cb\u003e7 Dispersion Decay 89\u003c\/b\u003e\u003c\/p\u003e \u003cp\u003e21 Proof of dispersion decay 90\u003c\/p\u003e \u003cp\u003e22 Low energy asymptotics 92\u003c\/p\u003e \u003cp\u003e\u003cb\u003e8 Scattering Theory and Spectral Resolution 97\u003c\/b\u003e\u003c\/p\u003e \u003cp\u003e23 Scattering theory 97\u003c\/p\u003e \u003cp\u003e23.1 Asymptotic completeness 97\u003c\/p\u003e \u003cp\u003e23.2 Wave and scattering operators 99\u003c\/p\u003e \u003cp\u003e23.3 Intertwining and commutation relations 99\u003c\/p\u003e \u003cp\u003e24 Spectral resolution 101\u003c\/p\u003e \u003cp\u003e24.1 Spectral resolution for the Schrödinger operator 101\u003c\/p\u003e \u003cp\u003e24.2 Diagonalization of scattering operator 101\u003c\/p\u003e \u003cp\u003e25 T Operator and SMatrix 1003\u003c\/p\u003e \u003cp\u003e\u003cb\u003e9 Scattering Cross Section 111\u003c\/b\u003e\u003c\/p\u003e \u003cp\u003e26 Introduction 111\u003c\/p\u003e \u003cp\u003e27 Main results 117\u003c\/p\u003e \u003cp\u003e28 Limiting Amplitude Principle 120\u003c\/p\u003e \u003cp\u003e29 Spherical waves 121\u003c\/p\u003e \u003cp\u003e30 Plane wave limit 125\u003c\/p\u003e \u003cp\u003e31 Convergence of flux 127\u003c\/p\u003e \u003cp\u003e32 Long range asymptotics 128\u003c\/p\u003e \u003cp\u003e33 Cross section 131\u003c\/p\u003e \u003cp\u003e\u003cb\u003e10 Klein-Gordon Equation 133\u003c\/b\u003e\u003c\/p\u003e \u003cp\u003e35 Introduction 134\u003c\/p\u003e \u003cp\u003e36 Free Klein-Gordon equation 137\u003c\/p\u003e \u003cp\u003e36.1 Dispersion decay 137\u003c\/p\u003e \u003cp\u003e36.2 Spectral properties 139\u003c\/p\u003e \u003cp\u003e37 Perturbed Klein-Gordon equation 143\u003c\/p\u003e \u003cp\u003e37.1 Spectral properties 143\u003c\/p\u003e \u003cp\u003e37.2 Dispersion decay 145\u003c\/p\u003e \u003cp\u003e38 Asymptotic completeness 149\u003c\/p\u003e \u003cp\u003e\u003cb\u003e11 Wave equation 151\u003c\/b\u003e\u003c\/p\u003e \u003cp\u003e39 Introduction 152\u003c\/p\u003e \u003cp\u003e40 Free wave equation 154\u003c\/p\u003e \u003cp\u003e40.1 Time-decay 154\u003c\/p\u003e \u003cp\u003e40.2 Spectral properties 155\u003c\/p\u003e \u003cp\u003e41 Perturbed wave equation 158\u003c\/p\u003e \u003cp\u003e41.1 Spectral properties 158\u003c\/p\u003e \u003cp\u003e41.2 Dispersion decay 160\u003c\/p\u003e \u003cp\u003e42 Asymptotic completeness 163\u003c\/p\u003e \u003cp\u003e43 Appendix: Sobolev embedding theorem 165\u003c\/p\u003e \u003cp\u003eReferences 167\u003c\/p\u003e \u003cp\u003eIndex 172\u003c\/p\u003e \u003cp\u003e“The book is carefully written, features \\complete and streamlined proofs\", and some material, such as a novel justification of the \\limiting amplitude principle\", appears here for the first time.”  (\u003ci\u003eZentralblatt MATH\u003c\/i\u003e, 1 September 2015)\u003c\/p\u003e  \u003cp\u003e\u003cb\u003eALEXANDER KOMECH, PhD,\u003c\/b\u003e 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.\u003c\/p\u003e \u003cp\u003e\u003cb\u003eELENA KOPYLOVA, PhD,\u003c\/b\u003e 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.\u003c\/p\u003e  \u003cp\u003e\u003cb\u003eA simplified, yet rigorous treatment of scattering theory methods and their applications\u003c\/b\u003e\u003c\/p\u003e \u003cp\u003e\u003ci\u003eDispersion Decay and Scattering Theory\u003c\/i\u003e 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.\u003c\/p\u003e \u003cp\u003eThe 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.\u003c\/p\u003e \u003cp\u003e\u003ci\u003eDispersion Decay and Scattering Theory\u003c\/i\u003e 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.\u003c\/p\u003e","brand":"Wiley","offers":[{"title":"Default Title","offer_id":47989076263141,"sku":"NP9781118341827","price":116.95,"currency_code":"USD","in_stock":false}],"thumbnail_url":"\/\/cdn.shopify.com\/s\/files\/1\/1842\/7735\/files\/9781118341827.jpg?v=1761782696","url":"https:\/\/k12savings.com\/products\/dispersion-decay-and-scattering-theory-isbn-9781118341827","provider":"K12savings","version":"1.0","type":"link"}