A note on the three dimensional Dirac operator with zigzag type boundary conditions

Abstract

In this note the three dimensional Dirac operator $A_m$ with boundary conditions, which are the analogue of the two dimensional zigzag boundary conditions, is investigated. It is shown that $A_m$ is self-adjoint in $L^2(\Omega;\mathbb{C}^4)$ for any open set $\Omega \subset \mathbb{R}^3$ and its spectrum is described explicitly in terms of the spectrum of the Dirichlet Laplacian in $\Omega$. In particular, whenever the spectrum of the Dirichlet Laplacian is purely discrete, then also the spectrum of $A_m$ consists of discrete eigenvalues that accumulate at $\pm \infty$ and one additional eigenvalue of infinite multiplicity.