论文标题
物理学中普遍存在的线性连续方程的统一视角。第五部分：分解；在他们的频谱上界限；当操作员不是自我偶然时，他们的stieltjes积分表示
A unifying perspective on linear continuum equations prevalent in physics. Part V: resolvents; bounds on their spectrum; and their Stieltjes integral representations when the operator is not selfadjoint
论文作者
论文摘要
我们考虑以$ {\ bf a} =γ_1{\ bf bf b}γ_1$为单位的运算符的回报，其中$γ_1（{\ bf k}）$是一项投影，是在傅立叶空间中在本地起作用的，$ {\ bf b}（\ bf b}（\ bf x}）当人们想解决在第一部分，II，III和IV中调查的任何大型线性物理方程时，这种分解自然就会产生，这些方程可以在扩展的复合材料的抽象理论中重新构成问题。我们回顾了$ q^*$ - 凸操作员可以使用$ {\ bf a} $绑定频谱。然后，基于Cherkaev-Gibiansky的转型和随后的发展，我们重新重新提出了我们为non-Hermitian $ {\ bf b} $ A stieltjes type type type积分表示，用于分解$（z_0 {Z_0 {\ bf i} - {\ bf i} - {\ bf a}）。该表示形式在半平面$ \ re（e^{i \ vartheta} z_0）> c $中，其中$ \ vartheta $和$ c $如此，以至于$ c {\ bf i} - [e^{e^{i \ vartheta}正确定（和强制性）。
We consider resolvents of operators taking the form ${\bf A}=Γ_1{\bf B}Γ_1$ where $Γ_1({\bf k})$ is a projection that acts locally in Fourier space and ${\bf B}({\bf x})$ is an operator that acts locally in real space. Such resolvents arise naturally when one wants to solve any of the large class of linear physical equations surveyed in Parts I, II, III, and IV that can be reformulated as problems in the extended abstract theory of composites. We review how $Q^*$-convex operators can be used to bound the spectrum of ${\bf A}$. Then, based on the Cherkaev-Gibiansky transformation and subsequent developments, that we reformulate, we obtain for non-Hermitian ${\bf B}$ a Stieltjes type integral representation for the resolvent $(z_0{\bf I}-{\bf A})^{-1}$. The representation holds in the half plane $\Re(e^{i\vartheta}z_0)>c$, where $\vartheta$ and $c$ are such that $c{\bf I}-[e^{i\vartheta}{\bf B}+e^{-i\vartheta}{\bf B}^\dagger]$ is positive definite (and coercive).
