Spectral Analysis of Relativistic Operators.
Başlık:
Spectral Analysis of Relativistic Operators.
Yazar:
Balinsky, A.A.
ISBN:
9781848162198
Yazar Ek Girişi:
Fiziksel Tanımlama:
1 online resource (200 pages)
İçerik:
Contents -- Preface -- Chapter 1 Preliminaries -- 1.1 Linear operators -- 1.2 Quadratic forms -- 1.3 Spectra of self-adjoint operators -- 1.4 Compact operators -- 1.5 Fourier and Mellin transforms -- 1.6 Sobolev spaces -- 1.7 Inequalities -- 1.8 CLR and related inequalities -- 1.8.1 The SchrŁodinger operator -- 1.8.2 The magnetic SchrŁodinger operator -- 1.8.3 The quasi-relativistic SchrŁodinger operator -- 1.8.4 The magnetic quasi-relativistic SchrŁodinger operator -- 1.9 Lieb-Thirring inequalities -- Chapter 2 Operators -- 2.1 The Dirac operator -- 2.1.1 Partial wave decomposition -- 2.1.2 Spherically symmetric electric potentials -- 2.1.3 Matrix-valued potentials -- 2.2 The quasi-relativistic operator -- 2.2.1 The Kato inequality -- 2.2.2 The self-adjoint realisation -- 2.2.3 The range 1/2 < γ< 2/π for h0 -- 2.3 The Brown-Ravenhall operator -- 2.3.1 Coulomb potentials -- 2.3.2 Lower semi-boundedness -- 2.3.3 The Brown-Ravenhall operators bl,s -- 2.4 A unique continuation property -- Chapter 3 Spectra -- 3.1 The Dirac operator -- 3.1.1 Preliminary lemmas -- 3.1.2 The essential spectrum of D -- 3.1.3 Eigenvalues in (-1, 1) -- 3.1.4 Min-max characterisation -- 3.2 The quasi-relativistic operator -- 3.2.1 The essential spectrum -- 3.3 The Brown-Ravenhall operator -- 3.3.1 The essential spectrum -- 3.3.2 The virial theorem -- 3.4 The absence of embedded eigenvalues -- Chapter 4 Miscellany -- 4.1 Stability of matter -- 4.2 Eigenvalues of the operators DA and PA -- 4.2.1 Examples of zero modes -- 4.2.2 Chern-Simons actions for magnetic fields with zero modes -- 4.2.3 Quaternionic description of zero modes -- 4.3 Zero modes of DA and PA in Rn, n ≥ 2. -- 4.4 Decay rates of weak solutions of DQΨ = 0 -- 4.4.1 Reduction by inversion -- 4.4.2 Embeddings of Dirac-Sobolev spaces -- 4.4.3 Decay of zero modes.
4.5 Sobolev and CLR inequalities for Pauli operators -- 4.6 One-electron relativistic molecules -- 4.6.1 Quasi-relativistic model -- 4.6.2 The Brown-Ravenhall model -- 4.7 Stability of matter in magnetic fields -- 4.7.1 Non-relativistic matter -- 4.7.2 Relativistic matter -- Bibliography -- Index -- List of Symbols.
Özet:
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.
Notlar:
Electronic reproduction. Ann Arbor, Michigan : ProQuest Ebook Central, 2017. Available via World Wide Web. Access may be limited to ProQuest Ebook Central affiliated libraries.
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