Topics in the theory of Schrödinger operators:
Gespeichert in:
| Format: | Buch |
|---|---|
| Sprache: | English |
| Veröffentlicht: |
New Jersey [u.a.]
World Scientific
2004
|
| Schlagworte: | |
| Online-Zugang: | Inhaltsverzeichnis |
| Beschreibung: | XI, 275 S. graph. Darst. |
| ISBN: | 9812387978 |
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| 300 | |a XI, 275 S. |b graph. Darst. | ||
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Datensatz im Suchindex
| _version_ | 1804136423000899584 |
|---|---|
| adam_text | CONTENTS
Preface v
CHAPTER 1 INTRODUCTION 1
H. Araki
1 Schrodinger Operator 1
2 Scattering Theory 3
3 References 7
CHAPTER 2 TIME-PERIODIC SCHRODINGER
EQUATIONS 9
K. Yajima
1 Introduction 9
2 Existence and Uniqueness of Propagator 10
2.1 Autonomous case, self-adjointness 11
2.2 Nonautonomous case 13
2.2.1 Semigroup theory 13
2.2.2 The method of integral equations 14
3 Floquet Hamiltonian 19
4 Scattering Theory 24
4.1 RAGE theorem 24
4.2 Existence and completeness of wave operators 25
4.3 Scattering operator 30
4.4 Uniqueness of inverse scattering 31
5 Local Decay Property 33
5.1 Expansion for free Schrodinger equation 33
5.2 Threshold resonance, theorems 34
5.3 Threshold expansion of resolvent 35
vii
viii
5.4 Outline of proof of Theorem 5.3 37
6 Dispersive Estimates 38
6.1 Lp-Lq estimates and Strichartz estimates 38
6.2 Local smoothing estimate 45
7 Instability of Bound States 49
7.1 Resonance theory for time periodic systems 50
7.2 Instability of bound states I, analytic case 53
7.3 Fermi golden rule 56
7.4 Instability of bound states II, non-analytic case 60
Acknowledgements 65
Bibliography 65
CHAPTER 3 AN APPLICATION OF PHASE
SPACE TUNNELING TO MULTISTATE SCAT¬
TERING THEORY 71
1 Introduction - Main Theorem 71
2 Preliminaries 73
2.1 /l-pseudodifferential operators 73
2.2 FBI transform 75
2.3 Exponential weight estimates in phase space 76
3 Scattering Matrix 78
3.1 Construction of approximation systems: Aj (j = l,2) ... 79
3.2 Construction of identification map J 80
3.3 Wave operators W = W±(A,A0;J) 81
3.4 Spectral representation of A on / 83
3.5 Wave operators W| = W±(A, A; J*) 84
3.6 Representation formula of S2 (A) 86
4 Exponential Estimates - Proof of Main Theorem 87
Bibliography 90
CHAPTER 4 INVERSE SPECTRAL THEORY 93
H. Isozaki
Part I One-dimensional Problem 94
1 Inverse Eigenvalue Problem 94
1.1 Theorem of Borg-Levinson 94
1.2 Global structure of isospectral potentials 95
2 Isospectral Deformation 96
3 Inverse Scattering 98
3.1 Scattering problem 98
Contents ix
3.2 Spherically symmetric potentials 99
3.3 Gel fand-Levitan theory 100
3.4 Generalized sine transformation 102
3.5 The core of Gel fand-Levitan theory 103
3.6 What is the hidden mechanism? 104
Part II Multi-Dimensional Problem 106
4 n-Dimensional Borg-Levinson Theorem 106
5 Generalized Gel fand Problem 108
6 Kac Problem Ill
6.1 Isospectral manifolds 112
6.2 Spectral invariants 112
7 Overdeterminacy 113
7.1 High energy Born approximation 113
7.2 Time-dependent inverse scattering 115
7.3 Overdeterminacy 115
7.4 Inverse back scattering 116
Part III Inverse Scattering in n-Dimensions 116
8 Key Idea of Faddeev 116
8.1 Spectral representation 116
8.2 Higher dimensional Volterra operator 117
8.3 Faddeev s Green operator 118
9 Changing Green Operators 121
10 Direction Dependent Green Operators 123
11 Inverse Scattering at a Fixed Energy 126
11.1 Perturbed Green operators 126
11.2 Faddeev scattering amplitude 127
11.3 Inverse scattering at a fixed energy 128
11.4 Slowly decreasing potentials 129
12 Faddeev Theory 129
12.1 Exceptional points 129
12.2 Factorization of 5-matrix 130
12.3 Volterra operator 131
12.4 Gel fand-Levitan equation 132
13 9-Approach 133
14 Inverse Conductivity Problem 135
15 Other Applications 138
Bibliography 139
X
CHAPTER 5 ANALYSIS OF GROUND STATES
OF ATOMS INTERACTING WITH A QUAN¬
TIZED RADIATION FIELD 145
F. Hiroshima
1 Preliminaries 145
1.1 Perturbations of embedded eigenvalues 147
1.2 Positive temperatures 149
1.3 Organization 151
2 The Pauli-Fierz Model 152
2.1 Classical Hamiltonian in the Coulomb gauge 153
2.2 Fock representation 155
2.3 Second quantization 158
2.4 The Pauli-Fierz Hamiltonian 160
2.5 Examples and infrared singularities 161
3 The List of Problems 164
4 The Existence of Ground States 169
4.1 Ultraviolet cutoffs and domains 170
4.2 Self-adjointness for weak couplings 171
4.3 Positive spectral gap 171
4.4 Enhanced binding 184
4.5 Asymptotic fields 191
4.6 Scaling limits 194
4.7 Notes 195
5 Functional Integral Representations 198
5.1 Schrodinger representation 199
5.2 Decompositions of semigroups 202
5.3 Functional integral representation I 205
6 Self-Adjointness for Arbitrary eeK 210
6.1 Self-adjoint extensions 210
6.2 Invariant domains 213
6.3 Self-adjointness 218
6.4 Functional integral representation II 220
7 Multiplicity 221
7.1 Positivity improving semigroups and uniqueness of a ground
state 222
7.2 Degenerate ground states with singular potentials 227
7.3 Twofold degenerate ground states with spin 229
7.3.1 Fixed total momentum 230
Contents xi
7.3.2 External potentials 237
8 Ground State Energy, Charge Density and Effective Mass .... 238
8.1 Ground state energy 238
8.2 Charge density in ground states 239
8.3 Effective mass and mass renormalization 241
9 Gibbs Measures 244
9.1 Infinite volume Gibbs measures 244
9.2 Kolmogorov extension theorem 245
9.3 Expectation values 248
10 The Nelson Model 252
10.1 The Nelson Hamiltonian 252
10.2 Functional integral representations for the Nelson model . . 253
10.3 Superexponential decay and infrared singularities 256
10.4 Boson density 263
10.5 Notes 265
Acknowledgements 265
Bibliography 265
Index 273
|
| adam_txt |
CONTENTS
Preface v
CHAPTER 1 INTRODUCTION 1
H. Araki
1 Schrodinger Operator 1
2 Scattering Theory 3
3 References 7
CHAPTER 2 TIME-PERIODIC SCHRODINGER
EQUATIONS 9
K. Yajima
1 Introduction 9
2 Existence and Uniqueness of Propagator 10
2.1 Autonomous case, self-adjointness 11
2.2 Nonautonomous case 13
2.2.1 Semigroup theory 13
2.2.2 The method of integral equations 14
3 Floquet Hamiltonian 19
4 Scattering Theory 24
4.1 RAGE theorem 24
4.2 Existence and completeness of wave operators 25
4.3 Scattering operator 30
4.4 Uniqueness of inverse scattering 31
5 Local Decay Property 33
5.1 Expansion for free Schrodinger equation 33
5.2 Threshold resonance, theorems 34
5.3 Threshold expansion of resolvent 35
vii
viii
5.4 Outline of proof of Theorem 5.3 37
6 Dispersive Estimates 38
6.1 Lp-Lq estimates and Strichartz estimates 38
6.2 Local smoothing estimate 45
7 Instability of Bound States 49
7.1 Resonance theory for time periodic systems 50
7.2 Instability of bound states I, analytic case 53
7.3 Fermi golden rule 56
7.4 Instability of bound states II, non-analytic case 60
Acknowledgements 65
Bibliography 65
CHAPTER 3 AN APPLICATION OF PHASE
SPACE TUNNELING TO MULTISTATE SCAT¬
TERING THEORY 71
1 Introduction - Main Theorem 71
2 Preliminaries 73
2.1 /l-pseudodifferential operators 73
2.2 FBI transform 75
2.3 Exponential weight estimates in phase space 76
3 Scattering Matrix 78
3.1 Construction of approximation systems: Aj (j = l,2) . 79
3.2 Construction of identification map J 80
3.3 Wave operators W\ = W±(A,A0;J) 81
3.4 Spectral representation of A on / 83
3.5 Wave operators W| = W±(A, A; J*) 84
3.6 Representation formula of S2 (A) 86
4 Exponential Estimates - Proof of Main Theorem 87
Bibliography 90
CHAPTER 4 INVERSE SPECTRAL THEORY 93
H. Isozaki
Part I One-dimensional Problem 94
1 Inverse Eigenvalue Problem 94
1.1 Theorem of Borg-Levinson 94
1.2 Global structure of isospectral potentials 95
2 Isospectral Deformation 96
3 Inverse Scattering 98
3.1 Scattering problem 98
Contents ix
3.2 Spherically symmetric potentials 99
3.3 Gel'fand-Levitan theory 100
3.4 Generalized sine transformation 102
3.5 The core of Gel'fand-Levitan theory 103
3.6 What is the hidden mechanism? 104
Part II Multi-Dimensional Problem 106
4 n-Dimensional Borg-Levinson Theorem 106
5 Generalized Gel'fand Problem 108
6 Kac Problem Ill
6.1 Isospectral manifolds 112
6.2 Spectral invariants 112
7 Overdeterminacy 113
7.1 High energy Born approximation 113
7.2 Time-dependent inverse scattering 115
7.3 Overdeterminacy 115
7.4 Inverse back scattering 116
Part III Inverse Scattering in n-Dimensions 116
8 Key Idea of Faddeev 116
8.1 Spectral representation 116
8.2 Higher dimensional Volterra operator 117
8.3 Faddeev's Green operator 118
9 Changing Green Operators 121
10 Direction Dependent Green Operators 123
11 Inverse Scattering at a Fixed Energy 126
11.1 Perturbed Green operators 126
11.2 Faddeev scattering amplitude 127
11.3 Inverse scattering at a fixed energy 128
11.4 Slowly decreasing potentials 129
12 Faddeev Theory 129
12.1 Exceptional points 129
12.2 Factorization of 5-matrix 130
12.3 Volterra operator 131
12.4 Gel'fand-Levitan equation 132
13 9-Approach 133
14 Inverse Conductivity Problem 135
15 Other Applications 138
Bibliography 139
X
CHAPTER 5 ANALYSIS OF GROUND STATES
OF ATOMS INTERACTING WITH A QUAN¬
TIZED RADIATION FIELD 145
F. Hiroshima
1 Preliminaries 145
1.1 Perturbations of embedded eigenvalues 147
1.2 Positive temperatures 149
1.3 Organization 151
2 The Pauli-Fierz Model 152
2.1 Classical Hamiltonian in the Coulomb gauge 153
2.2 Fock representation 155
2.3 Second quantization 158
2.4 The Pauli-Fierz Hamiltonian 160
2.5 Examples and infrared singularities 161
3 The List of Problems 164
4 The Existence of Ground States 169
4.1 Ultraviolet cutoffs and domains 170
4.2 Self-adjointness for weak couplings 171
4.3 Positive spectral gap 171
4.4 Enhanced binding 184
4.5 Asymptotic fields 191
4.6 Scaling limits 194
4.7 Notes 195
5 Functional Integral Representations 198
5.1 Schrodinger representation 199
5.2 Decompositions of semigroups 202
5.3 Functional integral representation I 205
6 Self-Adjointness for Arbitrary eeK 210
6.1 Self-adjoint extensions 210
6.2 Invariant domains 213
6.3 Self-adjointness 218
6.4 Functional integral representation II 220
7 Multiplicity 221
7.1 Positivity improving semigroups and uniqueness of a ground
state 222
7.2 Degenerate ground states with singular potentials 227
7.3 Twofold degenerate ground states with spin 229
7.3.1 Fixed total momentum 230
Contents xi
7.3.2 External potentials 237
8 Ground State Energy, Charge Density and Effective Mass . 238
8.1 Ground state energy 238
8.2 Charge density in ground states 239
8.3 Effective mass and mass renormalization 241
9 Gibbs Measures 244
9.1 Infinite volume Gibbs measures 244
9.2 Kolmogorov extension theorem 245
9.3 Expectation values 248
10 The Nelson Model 252
10.1 The Nelson Hamiltonian 252
10.2 Functional integral representations for the Nelson model . . 253
10.3 Superexponential decay and infrared singularities 256
10.4 Boson density 263
10.5 Notes 265
Acknowledgements 265
Bibliography 265
Index 273 |
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| illustrated | Illustrated |
| index_date | 2024-07-02T17:06:53Z |
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| spellingShingle | Topics in the theory of Schrödinger operators Kwantummechanica gtt Schrödingervergelijking gtt Quantentheorie Quantum theory Schrödinger operator Inverse Spektraltheorie (DE-588)4314037-3 gnd Hamilton-Operator (DE-588)4072278-8 gnd Operatortheorie (DE-588)4075665-8 gnd Differentialoperator (DE-588)4012251-7 gnd |
| subject_GND | (DE-588)4314037-3 (DE-588)4072278-8 (DE-588)4075665-8 (DE-588)4012251-7 |
| title | Topics in the theory of Schrödinger operators |
| title_alt | Schrödinger operators |
| title_auth | Topics in the theory of Schrödinger operators |
| title_exact_search | Topics in the theory of Schrödinger operators |
| title_exact_search_txtP | Topics in the theory of Schrödinger operators |
| title_full | Topics in the theory of Schrödinger operators ed. by Huzihiro Araki ... |
| title_fullStr | Topics in the theory of Schrödinger operators ed. by Huzihiro Araki ... |
| title_full_unstemmed | Topics in the theory of Schrödinger operators ed. by Huzihiro Araki ... |
| title_short | Topics in the theory of Schrödinger operators |
| title_sort | topics in the theory of schrodinger operators |
| topic | Kwantummechanica gtt Schrödingervergelijking gtt Quantentheorie Quantum theory Schrödinger operator Inverse Spektraltheorie (DE-588)4314037-3 gnd Hamilton-Operator (DE-588)4072278-8 gnd Operatortheorie (DE-588)4075665-8 gnd Differentialoperator (DE-588)4012251-7 gnd |
| topic_facet | Kwantummechanica Schrödingervergelijking Quantentheorie Quantum theory Schrödinger operator Inverse Spektraltheorie Hamilton-Operator Operatortheorie Differentialoperator |
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