Dispersion Decay and Scattering Theory (PDF)

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 lambda > 0 77 19.2 The case lambda = 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

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.

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