Dispersion Decay and Scattering Theory

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 10.3 Perturbation arguments 33 10.4 Continuous spectrum 35 10.5 Some improvements 36 4 Spectral Theory 37 11 Spectral representation 37 11.1 Inversion of Fourier-Laplace transform 37 11.2 Stationary Schrödinger equation 39 11.3 Spectral representation 39 11.4 Commutation relation 40 12 Analyticity of resolvent 41 13 Gohberg-Bleher theorem 43 14 Meromorphic continuation of resolvent 47 15 Absence of positive eigenvalues 50 15.1 Decay of eigenfunctions 50 15.2 Carleman estimates 54 15.3 Proof of Kato Theorem 56 5 High Energy Decay of Resolvent 59 16 High energy decay of free resolvent 59 16.1 Resolvent estimates 60 16.2 Decay of free resolvent 64 16.3 Decay of derivatives 65 17 High energy decay of perturbed resolvent 67 6 Limiting Absorption Principle 71 18 Free resolvent 71 19 Perturbed resolvent 77 19.1 The case " > 0 77 19.2 The case " = 0 78 20 Decay of eigenfunctions 81 20.1 Zero trace 81 20.2 Division problem 83 20.3 Negative eigenvalues 86 20.4 Appendix A: Sobolev Trace Theorem 86 20.5 Appendix B: SokhotskyPlemelj formula 87 7 Dispersion Decay 89 21 Proof of dispersion decay 90 22 Low energy asymptotics 92 8 Scattering Theory and Spectral Resolution 97 23 Scattering theory 97 23.1 Asymptotic completeness 97 23.2 Wave and scattering operators 99 23.3 Intertwining and commutation relations 99 24 Spectral resolution 101 24.1 Spectral resolution for the Schrödinger operator 101 24.2 Diagonalization of scattering operator 101 25 T Operator and SMatrix 1003 9 Scattering Cross Section 111 26 Introduction 111 27 Main results 117 28 Limiting Amplitude Principle 120 29 Spherical waves 121 30 Plane wave limit 125 31 Convergence of flux 127 32 Long range asymptotics 128 33 Cross section 131 10 Klein-Gordon Equation 133 35 Introduction 134 36 Free Klein-Gordon equation 137 36.1 Dispersion decay 137 36.2 Spectral properties 139 37 Perturbed Klein-Gordon equation 143 37.1 Spectral properties 143 37.2 Dispersion decay 145 38 Asymptotic completeness 149 11 Wave equation 151 39 Introduction 152 40 Free wave equation 154 40.1 Time-decay 154 40.2 Spectral properties 155 41 Perturbed wave equation 158 41.1 Spectral properties 158 41.2 Dispersion decay 160 42 Asymptotic completeness 163 43 Appendix: Sobolev embedding theorem 165 References 167 Index 172