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

2020/06/30 by Holzmann, Markus · 1 citation
#Analysis of PDEs (math.AP) #FOS: Mathematics #FOS: Physical sciences #Mathematical Physics (math-ph) #Spectral Theory (math.SP)

paper · doi:10.48550/arxiv.2006.16739

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 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 A_m consists of discrete eigenvalues that accumulate at ± ∞ and one additional eigenvalue of infinite multiplicity.

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