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 boundary conditions as of Robin type. 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 an exterior domain, and corresponding trace formulas.

中文翻译：

---

R 3 中域上的自伴狄拉克算子。


在本文中， \(L^2(\Omega ; \mathbb {C}^4)\)中一系列自伴狄拉克算子的光谱和散射特性，其中\(\Omega \subset \mathbb {R }^3\)是有界域或无界域，具有紧凑的\(C^2\)光滑边界，以系统的方式进行研究。这些算子可以被视为具有罗宾类型边界条件的拉普拉斯算子的自然相对论对应物。我们的方法基于来自对称算子可拓理论的抽象边界三重技术，以及对出现在 Krein 型解析公式中的某些类别（边界）积分算子的深入研究。对这个公式中扰动项的分析导致了对光谱和伯曼-施温格原理的描述，对\(\Omega \)为外域情况下的散射特性的定性理解，以及相应的迹公式。