ABSTRACT

We consider the 3-D Dirac operator with variable regular magnetic and electrostatic potentials, and singular potentials (1) ${\mathfrak{D}}_{\mathit{A},\mathrm{\Phi },Q_{\sin }}\mathit{u}\left(x\right)=\left({\mathfrak{D}}_{\mathit{A},\mathrm{\Phi }}+Q_{\sin }\right)\mathit{u}\left(x\right),x\in {\mathbb{R}}^3$(1) where (2) ${\mathfrak{D}}_{\mathit{A},\mathrm{\Phi }}=\sum _{j=1}^3{\alpha }_j\left(i{\mathrm{\partial }}_{x_j}+A_j\left(x\right)\right)+{\alpha }_{0}m+\mathrm{\Phi }\left(x\right)I_4,$(2) $Q_{\sin }=\mathrm{\Gamma }\left(s\right){\delta }_{\mathrm{\Sigma }}$ is the singular potential with $\mathrm{\Gamma }\left(s\right)={\left({\mathrm{\Gamma }}_{ij}\left(s\right)\right)}_{i,j=1}^4$ being a $4\times 4$ matrix and ${\delta }_{\mathrm{\Sigma }}$ is the delta-function with support on a surface $\mathrm{\Sigma }\subset {\mathbb{R}}^3$ which divides ${\mathbb{R}}^3$ on two open domains ${\mathrm{\Omega }}_{\pm }$ with the common boundary Σ, $\mathit{u}$ is a vector-function on ${\mathbb{R}}^3$ with values in ${\mathbb{C}}^4,{\alpha }_j,j=0,1,2,3$ are the standard $4\times 4$ Dirac matrices. We associate with the formal Dirac operator ${\mathfrak{D}}_{\mathit{A},\mathrm{\Phi },Q_{\sin }}$ an unbounded operator $\mathcal{D}$ in $L^2({\mathbb{R}}^3,{\mathbb{C}}^4)$ generated by ${\mathfrak{D}}_{\mathit{A},\mathrm{\Phi }}$ with domain in $H^1({\mathrm{\Omega }}_{+},{\mathbb{C}}^4)\oplus H^1({\mathrm{\Omega }}_{-},{\mathbb{C}}^4)$ consisting of functions satisfying transmission conditions on Σ. We consider the self-adjointness of operator $\mathcal{D}$, its Fredholm properties, and the essential spectrum in the case if Σ is either a closed $C^2$-surface or an unbounded $C^2$-hypersurface with a regular behaviour at infinity.

As application we consider the electrostatic and Lorentz scalar δ-shell interactions.

摘要

我们考虑具有可变规则磁势和静电势以及奇异势的 3-D Dirac 算子(1)${\mathfrak{D}}_{\mathit{一种},\mathrm{\Phi },{问}_{罪}}\mathit{你}\left(X\right)=\left({\mathfrak{D}}_{\mathit{一种},\mathrm{\Phi }}+{问}_{罪}\right)\mathit{你}\left(X\right),X\in {\mathbb{R}}^3$(1)哪里(2)${\mathfrak{D}}_{\mathit{一种},\mathrm{\Phi }}=\sum _{j=1}^3{\alpha }_j\left(\mathrm{一世}{\mathrm{\partial }}_{X_j}+{\mathrm{一种}}_j\left(X\right)\right)+{\alpha }_0米+\mathrm{\Phi }\left(X\right){\mathrm{一世}}_4,$(2) ${问}_{罪}=\mathrm{\Gamma }\left(s\right){\delta }_{\mathrm{\Sigma }}$是奇异势$\mathrm{\Gamma }\left(s\right)={\left({\mathrm{\Gamma }}_{\mathrm{一世}j}\left(s\right)\right)}_{\mathrm{一世},j=1}^4$成为一个$4\times 4$矩阵和${\delta }_{\mathrm{\Sigma }}$是表面上支持的 delta 函数$\mathrm{\Sigma }\subset {\mathbb{R}}^3$哪个划分${\mathbb{R}}^3$在两个开放域上${\mathrm{\Omega }}_{\pm }$与公共边界 Σ，$\mathit{你}$是一个向量函数${\mathbb{R}}^3$与值${\mathbb{C}}^4,{\alpha }_j,j=0,1,2,3$是标准$4\times 4$狄拉克矩阵。我们与正式的狄拉克算子相关联${\mathfrak{D}}_{\mathit{一种},\mathrm{\Phi },{问}_{罪}}$无界算子$\mathcal{D}$在${\mathrm{大号}}^2({\mathbb{R}}^3,{\mathbb{C}}^4)$由产生${\mathfrak{D}}_{\mathit{一种},\mathrm{\Phi }}$域在$H^1({\mathrm{\Omega }}_{+},{\mathbb{C}}^4)\oplus H^1({\mathrm{\Omega }}_{-},{\mathbb{C}}^4)$由满足 Σ 上的传输条件的函数组成。我们考虑算子的自伴随性$\mathcal{D}$，它的 Fredholm 属性，以及如果 Σ 是闭合的情况下的本质谱$C^2$-表面或无界$C^2$-在无穷远处具有规律行为的超曲面。

作为应用，我们考虑静电和洛伦兹标量δ -壳相互作用。