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.
Highlights
In the recent years Dirac operators with boundary conditions, which make them selfadjoint, gained a lot of attention
The two dimensional zigzag boundary conditions have a physical relevance, as they appear in the description of graphene quantum dots, when a lattice in this quantum dot is terminated and the direction of the boundary is perpendicular to the bonds [9]
It was shown in [16] that the two dimensional Dirac operator with these zigzag boundary conditions is self-adjoint on a domain which is in general not contained in H 1( ) and that for any bounded domain zero is an eigenvalue with infinite multiplicity
Summary
In the recent years Dirac operators with boundary conditions, which make them selfadjoint, gained a lot of attention. This article is part of the topical collection “Recent Developments in Operator Theory - Contributions in Honor of H.S.V. de Snoo” edited by Jussi Behrndt and Seppo Hassi
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