论文标题
具有锯齿形类型边界条件的三维狄拉克运算符上的注释
A note on the three dimensional Dirac operator with zigzag type boundary conditions
论文作者
论文摘要
在本说明中，研究了三维狄拉克运算符$ a_m $具有边界条件的$ a_m $，这是两个维度锯齿形边界条件的类似物。结果表明，对于任何开放式设置$ l^2（ω; \ mathbb {c}^4）$的$ a_m $对于任何开放式集合$ω\ subset \ subset \ mathbb {r}^3 $及其频谱的频谱以$ω$ $ω$的dirichlet laplacian的频谱进行明确描述。特别是，每当dirichlet laplacian的频谱纯粹是离散的时，$ a_m $的光谱也会由离散的特征值组成，这些特征值以$ \ pm \ pm \ infty $和一个额外的无限多样性特征值组成。
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 $L^2(Ω;\mathbb{C}^4)$ for any open set $Ω\subset \mathbb{R}^3$ 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 $\pm \infty$ and one additional eigenvalue of infinite multiplicity.
