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

TL;DR

This paper systematically studies the spectral and scattering properties of self-adjoint Dirac operators in three-dimensional domains, introducing new boundary techniques and analyzing the MIT bag model within a rigorous mathematical framework.

Contribution

It develops a comprehensive boundary triple approach for Dirac operators with Robin-like conditions, including the MIT bag model, and derives spectral, scattering, and trace formulas.

Findings

Spectral characterization of Dirac operators on various domains.

Scattering properties and Birman-Schwinger principle established.

Trace formulas derived for the operators.

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.