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

TL;DR

This paper analyzes a three-dimensional Dirac operator with boundary conditions similar to zigzag boundaries, showing it is self-adjoint and explicitly describing its spectrum in relation to the Dirichlet Laplacian.

Contribution

It introduces and investigates the 3D Dirac operator with zigzag-like boundary conditions, providing explicit spectral characterization and establishing self-adjointness.

Findings

The operator is self-adjoint in L^2 space.

Spectrum is explicitly described via the Dirichlet Laplacian.

Discrete eigenvalues accumulate at infinity with an additional infinite multiplicity eigenvalue.

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.