论文标题
Fredholm的财产和$ 3-D $ DIRAC运营商的基本频谱，具有常规和奇异的潜力
Fredholm property and essential spectrum of $3-D$ Dirac operators with regular and singular potentials
论文作者
论文摘要
我们考虑$ 3-d $ dirac运算符$ \ mathfrak {d} _ {\ boldsymbol {a}，φ，q _ {\ sin}} $具有可变的常规磁性和静电电位$ \ boldsymbol {a a} $，$ q _ $ q sin { \ mathbb {r}^{3} $，该$在两个开放域中划分$ \ mathbb {r}^{3} $ $ω_ {\ pm} $。我们与正式的狄拉克运算符$ \ mathfrak {d} _ {\ boldsymbol {a}，φ，q _ {\ sin}} $一个无限的操作员$ \ mathcal {d} _ {\ boldsymbol {\ boldsymbol { l^{2}（\ Mathbb {r}^{3}，\ Mathbb {C}^{4}）$由$ \ Mathfrak的常规部分生成$ h^{1}（ω_{+}，\ Mathbb {c}^{4}）\ oplus h^{1}（ω_ { - { - }，\ mathbb {c}^{4}^{4}^{4}）$ coptators $ nivessys $ nifferness $ nive $ c。 $ \ MATHCAL {d} _ {\ BOLDSYMBOL {a}，φ，q _ {\ sin}} $对于无界的$ C^{2} - $均匀的常规表面$σ，$和$ \ MATHCAL {d} d}的基本频谱， }} $如果$σ$是$ c^{2} $ - 圆锥出口到无限的表面。作为应用程序，我们考虑静电和Lorentz标量$δ_{σ} - $ shell互动在无界表面上的$σ。$。
We consider the $3-D$ Dirac operator $\mathfrak{D}_{\boldsymbol{A},Φ,Q_{\sin }}$ with variable regular magnetic and electrostatic potentials $ \boldsymbol{A}$,$Φ$ and with singular potentials $Q_{\sin }$ with support on a smooth unbounded surface $Σ\subset \mathbb{R}^{3}$ which divides $\mathbb{R}^{3}$ on two open domains $Ω_{\pm }$. We associate with the formal Dirac operator $\mathfrak{D}_{\boldsymbol{A},Φ,Q_{\sin }} $ an unbounded operator $\mathcal{D}_{\boldsymbol{A},Φ,Q_{\sin }}$ in $ L^{2}(\mathbb{R}^{3},\mathbb{C}^{4})$ generated by the regular part of $ \mathfrak{D}_{\boldsymbol{A},Φ,Q_{\sin }}$ with domain in $H^{1}(Ω_{+},\mathbb{C}^{4})\oplus H^{1}(Ω_{-},\mathbb{C}^{4})$ consisting of functions satisfying transmission conditions on $Σ.$ We consider the self-adjointness of operator $\mathcal{D}_{\boldsymbol{A},Φ,Q_{\sin }}$ for unbounded $C^{2}-$uniformly regular surfaces $Σ,$ and the essential spectrum of $\mathcal{D}_{\boldsymbol{A},Φ,Q_{\sin }}$ if $ Σ$ is a $C^{2}$-surfaces with conic exits to infinity. As application we consider the electrostatic and Lorentz scalar $δ_{Σ}-$shell interactions on unbounded surfaces $Σ.$
