论文标题
具有单数相互作用的狄拉克运算符的边界三元和WEYL功能
Boundary triples and Weyl functions for Dirac operators with singular interactions
论文作者
论文摘要
在本文中，我们开发了一种系统的方法来处理零dirac操作员$ a_ {η，τ，λ} $，带有单数静电，洛伦兹标量和异常的磁相互作用$η，τ，λ\ in \ mathbb {r} $，分别在$ \ toss in \ mathbb {r} $中支持。 $ \ mathbb {r}^2 $，以及$ \ mathbb {r}^3 $中的表面，基于边界三元及其相关的Weyl功能。首先，我们讨论了一维情况，该案例也是多维环境的动机。之后，在两个和三维情况下，我们构建了准，广义和普通边界的三元组及其Weyl功能，并提供了相关的Sobolev空间，跟踪定理以及积分操作员的映射特性的详细表征，这些属性在$ a_ {η，τ，λ} $中起着重要作用。我们朝着更粗糙的交互作用迈出了重大步骤，支持$σ$，并考虑一般紧凑的Lipschitz Hypersurfaces。我们得出了相互作用强度的条件，使操作员$ a_ {η，τ，λ} $是自我接合，获得krein型的分解公式，并表征必要的和离散的频谱。这些条件包括纯粹的洛伦兹标量和纯粹的非关键异常磁相互作用以及限制案例，后者在石墨烯的数学描述中具有重要的应用。使用某个普通的边界三倍，我们显示了$ a_ {η，τ，λ} $的自相关性，以在$σ$为$σ$为$ c^{\ infty} $的条件下，用于相互作用强度的任意组合（包括临界） - 平滑并得出光谱属性。特别是，在关键情况下，观察到操作员域中的Sobolev规则性丧失，并且观察到基本频谱的可能附加点。
In this article we develop a systematic approach to treat Dirac operators $A_{η, τ, λ}$ with singular electrostatic, Lorentz scalar, and anomalous magnetic interactions of strengths $η, τ, λ\in \mathbb{R}$, respectively, supported on points in $\mathbb{R}$, curves in $\mathbb{R}^2$, and surfaces in $\mathbb{R}^3$ that is based on boundary triples and their associated Weyl functions. First, we discuss the one-dimensional case which also serves as a motivation for the multidimensional setting. Afterwards, in the two and three-dimensional situation we construct quasi, generalized, and ordinary boundary triples and their Weyl functions, and provide a detailed characterization of the associated Sobolev spaces, trace theorems, and the mapping properties of integral operators which play an important role in the analysis of $A_{η, τ, λ}$. We make a substantial step towards more rough interaction supports $Σ$ and consider general compact Lipschitz hypersurfaces. We derive conditions for the interaction strengths such that the operators $A_{η, τ, λ}$ are self-adjoint, obtain a Krein-type resolvent formula, and characterize the essential and discrete spectrum. These conditions include purely Lorentz scalar and purely non-critical anomalous magnetic interactions as well as the confinement case, the latter having an important application in the mathematical description of graphene. Using a certain ordinary boundary triple, we show the self-adjointness of $A_{η, τ, λ}$ for arbitrary combinations of the interaction strengths (including critical ones) under the condition that $Σ$ is $C^{\infty}$-smooth and derive its spectral properties. In particular, in the critical case, a loss of Sobolev regularity in the operator domain and a possible additional point of the essential spectrum are observed.
