Dispersion Decay and Scattering Theory

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.

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 λ > 0 77

19.2 The case λ = 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