论文标题
MIT包模型的Poincaré-Steklov地图
A Poincaré-Steklov map for the MIT bag model
论文作者
论文摘要
本文的目的是介绍和研究与Dirac Operator相关的Poincaré-Steklov（PS）运营商$ d_m $与所谓的MIT袋边界条件。在域中$ω\ subset \ mathbb {r}^3 $，对于复杂的数字$ z $，对于$ u_z $，解决方案$（d_m-z）u_z = 0 $，关联的PS操作员将$γ__-U_z $的$ ud_z $的$ u_z $ undoction $ usive ud__ uspon upportion与$γ_+usγ_+usγ_ $ \partialΩ$和$（γ_- +γ_ +）= t _ {\partialΩ} $是$ \partialΩ$上的跟踪操作员。 在本文的第一部分中，我们表明PS运算符是零订单的伪差操作员，并给出其主要符号。在第二部分中，我们在质量$ m $大的情况下研究PS操作员，并且我们证明它适合$ 1/m $ - $ -PESEUDODIFFERCENTICTENTER EMALION的框架，并且我们得出了一些重要的属性，尤其是其半经典的主要符号。随后，我们应用这些结果来建立dirac操作员的krein型分解公式$ h_m = d_m+mβ1_ {\ m马理{r}^3 \ setMinus \overlineΩ} $，用于大块$ m> 0 $，就$ mit袋$ω$ω$ω$Ω而言。在其帮助下，显示了$ \ mathcal {o}（m^{ - 1}）$的收敛速率的大耦合收敛。
The purpose of this paper is to introduce and study Poincaré-Steklov (PS) operators associated to the Dirac operator $D_m$ with the so-called MIT bag boundary condition. In a domain $Ω\subset\mathbb{R}^3$, for a complex number $z$ and for $U_z$ a solution of $(D_m-z)U_z=0$, the associated PS operator maps the value of $Γ_- U_z$, the MIT bag boundary value of $U_z$, to $Γ_+ U_z$, where $Γ_\pm$ are projections along the boundary $\partialΩ$ and $(Γ_ - + Γ_+) = t_{\partialΩ}$ is the trace operator on $\partialΩ$. In the first part of this paper, we show that the PS operator is a zero-order pseudodifferential operator and give its principal symbol. In the second part, we study the PS operator when the mass $m$ is large, and we prove that it fits into the framework of $1/m$-pseudodifferential operators, and we derive some important properties, especially its semiclassical principal symbol. Subsequently, we apply these results to establish a Krein-type resolvent formula for the Dirac operator $H_M= D_m+ Mβ1_{\mathbb{R}^3\setminus\overlineΩ}$ for large masses $M>0$, in terms of the resolvent of the MIT bag operator on $Ω$. With its help, the large coupling convergence with a convergence rate of $\mathcal{O}(M^{-1})$ is shown.
