Over the last decade, there has been considerable interest and progress in determining the spectral properties of various operators that take relativistic effects into account, with important implications for mathematics and physics. Difficulties are encountered in many-particle problems due to the lack of semiboundedness of the Dirac operator, and this has led to the investigation of operators like those of Chandrasekhar-Herbst and Brown-Ravenhall, which are semibounded under appropriate circumstances. This book contains an up-to-date, comprehensive and self-contained analysis of the spectral properties of these operators, providing the tools for anyone working in this area. Another major feature is the work of the authors on zero modes, a topic which has important significance for the stability of matter and other physical problems. Up until now, these topics have been scattered throughout the literature, without a systematic and cohesive treatment. The book will report largely on the progress on these topics published since 1992.
Author(s): A. A. Balinsky, W. D. Evans
Publisher: Imperial College Press
Year: 2010
Language: English
Pages: 200
Tags: Математика;Функциональный анализ;Теория операторов;
Contents......Page 12
Preface......Page 8
1.1 Linear operators......Page 14
1.2 Quadratic forms......Page 16
1.3 Spectra of self-adjoint operators......Page 20
1.4 Compact operators......Page 21
1.5 Fourier and Mellin transforms......Page 22
1.6 Sobolev spaces......Page 28
1.7 Inequalities......Page 30
1.8 CLR and related inequalities......Page 33
1.8.3 The quasi-relativistic SchrŁodinger operator......Page 36
1.8.4 The magnetic quasi-relativistic SchrŁodinger operator......Page 37
1.9 Lieb-Thirring inequalities......Page 38
2.1 The Dirac operator......Page 40
2.1.1 Partial wave decomposition......Page 42
2.1.2 Spherically symmetric electric potentials......Page 47
2.1.3 Matrix-valued potentials......Page 54
2.2 The quasi-relativistic operator......Page 59
2.2.1 The Kato inequality......Page 60
2.2.2 The self-adjoint realisation......Page 66
2.2.3 The range 1/2 < γ< 2/π for h0......Page 70
2.3 The Brown-Ravenhall operator......Page 75
2.3.1 Coulomb potentials......Page 77
2.3.2 Lower semi-boundedness......Page 80
2.3.3 The Brown-Ravenhall operators bl,s......Page 90
2.4 A unique continuation property......Page 94
3.1.1 Preliminary lemmas......Page 98
3.1.2 The essential spectrum of D......Page 100
3.1.3 Eigenvalues in (-1, 1)......Page 101
3.2.1 The essential spectrum......Page 105
3.3.1 The essential spectrum......Page 108
3.3.2 The virial theorem......Page 109
3.4 The absence of embedded eigenvalues......Page 114
4.1 Stability of matter......Page 122
4.2.1 Examples of zero modes......Page 126
4.2.2 Chern-Simons actions for magnetic fields with zero modes......Page 133
4.2.3 Quaternionic description of zero modes......Page 134
4.3 Zero modes of DA and PA in Rn, n ≥ 2.......Page 139
4.4 Decay rates of weak solutions of DQΨ = 0......Page 146
4.4.1 Reduction by inversion......Page 147
4.4.2 Embeddings of Dirac-Sobolev spaces......Page 149
4.4.3 Decay of zero modes......Page 154
4.5 Sobolev and CLR inequalities for Pauli operators......Page 155
4.6.1 Quasi-relativistic model......Page 162
4.6.2 The Brown-Ravenhall model......Page 166
4.7 Stability of matter in magnetic fields......Page 181
4.7.1 Non-relativistic matter......Page 182
4.7.2 Relativistic matter......Page 186
Bibliography......Page 190
Index......Page 196
List of Symbols......Page 198
