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Self-Adjoint Dirac Operators on Domains in
Jussi Behrndt1, Markus Holzmann1, Albert Mas2
1Institut für Angewandte Mathematik, Technische Universität Graz, Steyrergasse 30, 8010 Graz, Austria.
Annales Henri Poincare
|August 9, 2020
Summary
This study systematically analyzes self-adjoint Dirac operators, exploring their spectral and scattering properties. The findings offer insights into relativistic quantum mechanics and boundary value problems.
Area of Science:
- Mathematical Physics
- Quantum Mechanics
- Spectral Theory
Background:
- Dirac operators are relativistic quantum mechanical operators.
- Understanding their spectral and scattering properties is crucial for various physics applications.
- Boundary conditions, particularly Robin-type, significantly influence operator behavior.
Purpose of the Study:
- To systematically study the spectral and scattering properties of self-adjoint Dirac operators.
- To establish a connection between these operators and relativistic counterparts of Laplacians.
- To analyze these properties for both bounded and unbounded domains with smooth boundaries.
Main Methods:
- Utilizing abstract boundary triple techniques from the extension theory of symmetric operators.
- Conducting a thorough study of specific classes of boundary integral operators.
- Employing a Krein-type resolvent formula and analyzing its perturbation term.
Main Results:
- A comprehensive description of the spectrum of the studied Dirac operators.
- Establishment of a Birman-Schwinger principle for these operators.
- Qualitative understanding of scattering properties, especially for exterior domains, and derivation of trace formulas.
Conclusions:
- The study provides a robust framework for analyzing Dirac operators with boundary conditions.
- The methods developed offer new tools for investigating spectral and scattering phenomena.
- The results contribute to a deeper understanding of relativistic quantum systems with boundaries.
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