Dispersion Decay and Scattering Theory Dispersion Decay and Scattering Theory Alexander Komech Vienna University and IITP RAS Elena Kopylova Vienna University and IITP RAS »WILEY A JOHN WILEY & SONS, INC., PUBLICATION Copyright © 2012 by John Wiley & Sons, Inc. All rights reserved Published by John Wiley & Sons, Inc., Hoboken, New Jersey Published simultaneously in Canada No part of this publication may be reproduced, stored in a retrieval system, or transmitted in any form or by any means, electronic, mechanical, photocopying, recording, scanning, or otherwise, except as permitted under Section 107 or 108 of the 1976 United States Copyright Act, without either the prior written permission of the Publisher, or authorization through payment of the appropriate per-copy fee to the Copyright Clearance Center, Inc., 222 Rosewood Drive, Danvers, MA 01923, (978) 750-8400, fax (978) 750-4470, or on the web at www.copyright.com. 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Kopylova, Elena, 1960-II. Title. QC174.26.W28K646 2012 530.12'4—dc23 2012007292 Printed in the United States of America. 10 9 8 7 6 5 4 3 21 CONTENTS IN BRIEF 1 Basic Concepts and Formulas 1 2 Nonstationary Schrödinger Equation 11 3 Stationary Schrödinger Equation 25 4 Spectral Theory 37 5 High Energy Decay of Resolvent 59 6 Limiting Absorption Principle 71 7 Dispersion Decay 89 8 Scattering Theory and Spectral Resolution 97 9 Scattering Cross Section 111 10 Klein-Gordon Equation 133 11 Wave equation 151 v CONTENTS List of Figures xi Foreword xiii Preface xv Acknowledgments xvii Introduction xix 1 Basic Concepts and Formulas 1 1 Distributions and Fourier transform 1 2 Functional spaces 3 2.1 Sobolev spaces 3 2.2 Agmon-Sobolev weighted spaces 4 2.3 Operator-valued functions 5 3 Free propagator 6 3.1 Fourier transform 6 3.2 Gaussian integrals 8 2 Nonstationary Schrodinger Equation 11 4 Definition of solution 11 vii viii CONTENTS 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 3 2 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 5 6 CONTENTS IX 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 ThecaseA>0 77 19.2 ThecaseA = 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: Sokhotsky-Plemelj 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 Schrodinger operator 101 24.2 Diagonalization of scattering operator 101 25 T-Operator and 5-Matrix 103 9 Scattering Cross Section 111 26 Introduction 111 27 Main results 117 X CONTENTS 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 34 Introduction 134 35 Free Klein-Gordon equation 137 35.1 Dispersion decay 137 35.2 Spectral properties 139 36 Perturbed Klein-Gordon equation 143 36.1 Spectral properties 143 36.2 Dispersion decay 145 37 Asymptotic completeness 149 11 Wave equation 151 38 Introduction 152 39 Free wave equation 154 39.1 Time decay 154 39.2 Spectral properties 155 40 Perturbed wave equation 158 40.1 Spectral properties 158 40.2 Dispersion decay 160 41 Asymptotic completeness 163 42 Appendix: Sobolev Embedding Theorem 165 References 167 Index 173 LIST OF FIGURES 1.1 Scattering and wave operators for ψ(0) e X. xxiii c 2 1.1 The function e~z decays in gray sectors. 9 2.1 Function (4.6). 12 3.1 The contour dist (ω, R+) = p. 27 3.2 Application of the Cauchy Residue Theorem. 29 3.3 The contour dist (ω, [Vb, oo)) = p. 33 3.4 The cutoff function. 36 4.1 The gray half-plane is the region of analyticity of ψ(ω). 38 4.2 The contours of integration. 41 4.3 Analytic continuation. 44 4.4 The contour C. 48 e 6.1 The cutoff function. 83 6.2 The contour of integration. 88 xi XÜ LIST OF FIGURES 7.1 The cutoff function. 90 9.1 Incident flux and scatterer. 112 9.2 Incident plane wave and outgoing spherical wave. 113 9.3 Incident and outgoing spherical waves. 118 10.1 The cutoff function. 138 10.2 The spectral gap and the continuous spectrum. 141 10.3 The cutoff functions. 148