Spectral analysis of relativistic operators - Balinsky, A. A. (Imperial College Press, 2011)
Master and use copy. Digital master created according to Benchmark for Faithful Digital Reproductions of Monographs and Serials, Version 1. Digital Library Federation, December 2002
Includes bibliographical references (pages 177-182) and index
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. 1.9. Lieb-Thirring inequalities -- 2. Operators. 2.1. The Dirac operator. 2.2. The quasi-relativistic operator. 2.3. The Brown-Ravenhall operator. 2.4. A unique continuation property -- 3. Spectra. 3.1. The Dirac operator. 3.2. The quasi-relativistic operator. 3.3. The Brown-Ravenhall operator. 3.4. The absence of embedded eigenvalues -- 4. Miscellany. 4.1. Stability of matter. 4.2. Eigenvalues of the operators D[symbol] and D[symbol]. 4.3. Zero modes of D[symbol] and P[symbol] in R[symbol], n[symbol]2. 4.4. Decay rates of weak solutions of [symbol]=0. 4.5. Sobolev and CLR inequalities for Pauli operators. 4.6. One-electron relativistic molecules. 4.7. Stability of matter in magnetic fields
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