Self-adjoint Dirac operators on domains in ${\mathbb{R}}^3$

Abstract

In this paper the spectral and scattering properties of a family of self-adjoint Dirac operators in $L^2(\Omega ;{\mathbb{C}}^4)$, where $\Omega \subset {\mathbb{R}}^3$ is either a bounded or an unbounded domain with a compact $C^2$-smooth boundary, are studied in a systematic way. These operators can be viewed as the natural relativistic counterpart of Laplacians with Robin boundary conditions. Among the Dirac operators treated here is also the so-called MIT bag operator, which has been used by physicists and more recently was discussed in the mathematical literature. Our approach is based on abstract boundary triple techniques from extension theory of symmetric operators and a thorough study of certain classes of (boundary) integral operators, that appear in a Krein-type resolvent formula. The analysis of the perturbation term in this formula leads to a description of the spectrum and a Birman-Schwinger principle, a qualitative understanding of the scattering properties in the case that $\Omega$ is unbounded, and corresponding trace formulas.

arXiv comment48 pages

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Cited by in corpus (9)

• General $\delta$-shell interactions for the two-dimensional Dirac operator: self-adjointness and approximation
• Spectral Properties of the Dirac Operator coupled with $\delta$-Shell Interactions
• Spectral Analysis of Dirac Operators with delta interactions supported on the boundaries of rough domains
• Eigenvalue curves for generalized MIT bag models
• Boundary problems for three-dimensional Dirac operators and generalized MIT bag models for unbounded domains
• Dirac operator spectrum in tubes and layers with a zigzag type boundary
• Convergence of generalized MIT bag models to Dirac operators with zigzag boundary conditions
• Detecting screens modeled by Schrödinger operators that generate $C_0$ contraction semigroups
• A Poincaré-Steklov map for the MIT bag model