Dispersion Decay and Scattering Theory

192 Pages·2012·3.854 MB·English

by  Alexander Komech, Elena Kopylova(auth.)

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
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Copyright © 2012 by John Wiley & Sons, Inc. All rights reserved
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Library of Congress Cataloging-in-Publication Data:
Komech,A. I., 1946-
Dispersion decay and scattering theory / Alexander Komech, Elena Kopylova.
p. cm.
Includes bibliographical references and index.
ISBN 978-1-118-34182-7 (cloth)
1. Klein-Gordon equation. 2. Spectral theory (Mathematics) 3. Scattering (Mathematics) I. 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