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.

“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.”  (Zentralblatt MATH, 1 September 2015)

• List of Figures xiii Foreword xvPreface xviiAcknowledgments xixIntroduction xxi1 Basic Concepts and Formulas 11 Distributions and Fourier transform 12 Functional spaces 32.1 Sobolev spaces 32.2 AgmonSobolev weighted spaces 42.3 Operatorvalued functions 53 Free propagator 63.1 Fourier transform 63.2 Gaussian integrals 82 Nonstationary Schrödinger Equation 114 Definition of solution 115 Schrödinger operator 145.1 A priori estimate 145.2 Hermitian symmetry 146 Dynamics for free Schrödinger equation 157 Perturbed Schrödinger equation 177.1 Reduction to integral equation 177.2 Contraction mapping 197.3 Unitarity and energy conservation 208 Wave and scattering operators 228.1 Möller wave operators. Cook method 228.2 Scattering operator 238.3 Intertwining identities 243 Stationary Schrödinger Equation 259 Free resolvent 259.1 General properties 259.2 Integral representation 2810 Perturbed resolvent 3110.1 Reduction to compact perturbation 3110.2 Fredholm Theorem 3210.3 Perturbation arguments 3310.4 Continuous spectrum 3510.5 Some improvements 364 Spectral Theory 3711 Spectral representation 3711.1 Inversion of Fourier-Laplace transform 3711.2 Stationary Schrödinger equation 3911.3 Spectral representation 3911.4 Commutation relation 4012 Analyticity of resolvent 4113 Gohberg-Bleher theorem 4314 Meromorphic continuation of resolvent 4715 Absence of positive eigenvalues 5015.1 Decay of eigenfunctions 5015.2 Carleman estimates 5415.3 Proof of Kato Theorem 565 High Energy Decay of Resolvent 5916 High energy decay of free resolvent 5916.1 Resolvent estimates 6016.2 Decay of free resolvent 6416.3 Decay of derivatives 6517 High energy decay of perturbed resolvent 676 Limiting Absorption Principle 7118 Free resolvent 7119 Perturbed resolvent 7719.1 The case λ > 0 7719.2 The case λ = 0 7820 Decay of eigenfunctions 8120.1 Zero trace 8120.2 Division problem 8320.3 Negative eigenvalues 8620.4 Appendix A: Sobolev Trace Theorem 8620.5 Appendix B: SokhotskyPlemelj formula 877 Dispersion Decay 8921 Proof of dispersion decay 9022 Low energy asymptotics 928 Scattering Theory and Spectral Resolution 9723 Scattering theory 9723.1 Asymptotic completeness 9723.2 Wave and scattering operators 9923.3 Intertwining and commutation relations 9924 Spectral resolution 10124.1 Spectral resolution for the Schrödinger operator 10124.2 Diagonalization of scattering operator 10125 T Operator and SMatrix 10039 Scattering Cross Section 11126 Introduction 11127 Main results 11728 Limiting Amplitude Principle 12029 Spherical waves 12130 Plane wave limit 12531 Convergence of flux 12732 Long range asymptotics 12833 Cross section 13110 Klein-Gordon Equation 13335 Introduction 13436 Free Klein-Gordon equation 13736.1 Dispersion decay 13736.2 Spectral properties 13937 Perturbed Klein-Gordon equation 14337.1 Spectral properties 14337.2 Dispersion decay 14538 Asymptotic completeness 14911 Wave equation 15139 Introduction 15240 Free wave equation 15440.1 Time-decay 15440.2 Spectral properties 15541 Perturbed wave equation 15841.1 Spectral properties 15841.2 Dispersion decay 16042 Asymptotic completeness 16343 Appendix: Sobolev embedding theorem 165References 167Index 172

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.

“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.”  (Zentralblatt MATH, 1 September 2015)

• List of Figures xiii Foreword xvPreface xviiAcknowledgments xixIntroduction xxi1 Basic Concepts and Formulas 11 Distributions and Fourier transform 12 Functional spaces 32.1 Sobolev spaces 32.2 AgmonSobolev weighted spaces 42.3 Operatorvalued functions 53 Free propagator 63.1 Fourier transform 63.2 Gaussian integrals 82 Nonstationary Schrödinger Equation 114 Definition of solution 115 Schrödinger operator 145.1 A priori estimate 145.2 Hermitian symmetry 146 Dynamics for free Schrödinger equation 157 Perturbed Schrödinger equation 177.1 Reduction to integral equation 177.2 Contraction mapping 197.3 Unitarity and energy conservation 208 Wave and scattering operators 228.1 Möller wave operators. Cook method 228.2 Scattering operator 238.3 Intertwining identities 243 Stationary Schrödinger Equation 259 Free resolvent 259.1 General properties 259.2 Integral representation 2810 Perturbed resolvent 3110.1 Reduction to compact perturbation 3110.2 Fredholm Theorem 3210.3 Perturbation arguments 3310.4 Continuous spectrum 3510.5 Some improvements 364 Spectral Theory 3711 Spectral representation 3711.1 Inversion of Fourier-Laplace transform 3711.2 Stationary Schrödinger equation 3911.3 Spectral representation 3911.4 Commutation relation 4012 Analyticity of resolvent 4113 Gohberg-Bleher theorem 4314 Meromorphic continuation of resolvent 4715 Absence of positive eigenvalues 5015.1 Decay of eigenfunctions 5015.2 Carleman estimates 5415.3 Proof of Kato Theorem 565 High Energy Decay of Resolvent 5916 High energy decay of free resolvent 5916.1 Resolvent estimates 6016.2 Decay of free resolvent 6416.3 Decay of derivatives 6517 High energy decay of perturbed resolvent 676 Limiting Absorption Principle 7118 Free resolvent 7119 Perturbed resolvent 7719.1 The case λ > 0 7719.2 The case λ = 0 7820 Decay of eigenfunctions 8120.1 Zero trace 8120.2 Division problem 8320.3 Negative eigenvalues 8620.4 Appendix A: Sobolev Trace Theorem 8620.5 Appendix B: SokhotskyPlemelj formula 877 Dispersion Decay 8921 Proof of dispersion decay 9022 Low energy asymptotics 928 Scattering Theory and Spectral Resolution 9723 Scattering theory 9723.1 Asymptotic completeness 9723.2 Wave and scattering operators 9923.3 Intertwining and commutation relations 9924 Spectral resolution 10124.1 Spectral resolution for the Schrödinger operator 10124.2 Diagonalization of scattering operator 10125 T Operator and SMatrix 10039 Scattering Cross Section 11126 Introduction 11127 Main results 11728 Limiting Amplitude Principle 12029 Spherical waves 12130 Plane wave limit 12531 Convergence of flux 12732 Long range asymptotics 12833 Cross section 13110 Klein-Gordon Equation 13335 Introduction 13436 Free Klein-Gordon equation 13736.1 Dispersion decay 13736.2 Spectral properties 13937 Perturbed Klein-Gordon equation 14337.1 Spectral properties 14337.2 Dispersion decay 14538 Asymptotic completeness 14911 Wave equation 15139 Introduction 15240 Free wave equation 15440.1 Time-decay 15440.2 Spectral properties 15541 Perturbed wave equation 15841.1 Spectral properties 15841.2 Dispersion decay 16042 Asymptotic completeness 16343 Appendix: Sobolev embedding theorem 165References 167Index 172

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.

“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.”  (Zentralblatt MATH, 1 September 2015)

• List of Figures xiii Foreword xvPreface xviiAcknowledgments xixIntroduction xxi1 Basic Concepts and Formulas 11 Distributions and Fourier transform 12 Functional spaces 32.1 Sobolev spaces 32.2 AgmonSobolev weighted spaces 42.3 Operatorvalued functions 53 Free propagator 63.1 Fourier transform 63.2 Gaussian integrals 82 Nonstationary Schrödinger Equation 114 Definition of solution 115 Schrödinger operator 145.1 A priori estimate 145.2 Hermitian symmetry 146 Dynamics for free Schrödinger equation 157 Perturbed Schrödinger equation 177.1 Reduction to integral equation 177.2 Contraction mapping 197.3 Unitarity and energy conservation 208 Wave and scattering operators 228.1 Möller wave operators. Cook method 228.2 Scattering operator 238.3 Intertwining identities 243 Stationary Schrödinger Equation 259 Free resolvent 259.1 General properties 259.2 Integral representation 2810 Perturbed resolvent 3110.1 Reduction to compact perturbation 3110.2 Fredholm Theorem 3210.3 Perturbation arguments 3310.4 Continuous spectrum 3510.5 Some improvements 364 Spectral Theory 3711 Spectral representation 3711.1 Inversion of Fourier-Laplace transform 3711.2 Stationary Schrödinger equation 3911.3 Spectral representation 3911.4 Commutation relation 4012 Analyticity of resolvent 4113 Gohberg-Bleher theorem 4314 Meromorphic continuation of resolvent 4715 Absence of positive eigenvalues 5015.1 Decay of eigenfunctions 5015.2 Carleman estimates 5415.3 Proof of Kato Theorem 565 High Energy Decay of Resolvent 5916 High energy decay of free resolvent 5916.1 Resolvent estimates 6016.2 Decay of free resolvent 6416.3 Decay of derivatives 6517 High energy decay of perturbed resolvent 676 Limiting Absorption Principle 7118 Free resolvent 7119 Perturbed resolvent 7719.1 The case λ > 0 7719.2 The case λ = 0 7820 Decay of eigenfunctions 8120.1 Zero trace 8120.2 Division problem 8320.3 Negative eigenvalues 8620.4 Appendix A: Sobolev Trace Theorem 8620.5 Appendix B: SokhotskyPlemelj formula 877 Dispersion Decay 8921 Proof of dispersion decay 9022 Low energy asymptotics 928 Scattering Theory and Spectral Resolution 9723 Scattering theory 9723.1 Asymptotic completeness 9723.2 Wave and scattering operators 9923.3 Intertwining and commutation relations 9924 Spectral resolution 10124.1 Spectral resolution for the Schrödinger operator 10124.2 Diagonalization of scattering operator 10125 T Operator and SMatrix 10039 Scattering Cross Section 11126 Introduction 11127 Main results 11728 Limiting Amplitude Principle 12029 Spherical waves 12130 Plane wave limit 12531 Convergence of flux 12732 Long range asymptotics 12833 Cross section 13110 Klein-Gordon Equation 13335 Introduction 13436 Free Klein-Gordon equation 13736.1 Dispersion decay 13736.2 Spectral properties 13937 Perturbed Klein-Gordon equation 14337.1 Spectral properties 14337.2 Dispersion decay 14538 Asymptotic completeness 14911 Wave equation 15139 Introduction 15240 Free wave equation 15440.1 Time-decay 15440.2 Spectral properties 15541 Perturbed wave equation 15841.1 Spectral properties 15841.2 Dispersion decay 16042 Asymptotic completeness 16343 Appendix: Sobolev embedding theorem 165References 167Index 172