Self-Adjoint Dirac Operators on Domains in R 3

Abstract

In this paper, the spectral and scattering properties of a family of self-adjoint Dirac operators in $L^2(\Omega ;C^4)$ , where $\Omega \subset R^3$ 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.

Keywords: Primary 81Q10; Secondary 35Q40.