by A A Balinsky (Cardiff University, UK) & W D Evans (Cardiff University, UK)

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.

• Preliminaries on Operator Theory in Hilbert Spaces, Fourier Theory and Mellin Transforms
• The Dirac Operator with Coulomb Potential, Essential-Self-Adjointness and Spectrum
• The Pseudo-Relativistic Operator of Chandrasekhar–Herbst, the Herbst Analysis of Its Spectrum and Kato's Inequality
• Description of Brown–Ravenhall Operator, Proof of Positivity, Location of Its Essential Spectrum, a Virial Theorem and Embedded Eigenvalues
• Zero Modes of Weyl–Dirac and Pauli Operators