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

In this note the three dimensional Dirac operator Am with boundary conditions, which are the analogue of the two dimensional zigzag boundary conditions, is investigated. It is shown that Am is self-adjoint in L2(Ω;C4) for any open set Ω⊂R3 and its spectrum is described explicitly in terms of the spectrum of the Dirichlet Laplacian in Ω. In particular, whenever the spectrum of the Dirichlet Laplacian is purely discrete, then also the spectrum of Am consists of discrete eigenvalues that accumulate at ±∞ and one additional eigenvalue of infinite multiplicity.