Spectral flow inside essential spectrum II: resonance set and its structure

Spectral flow inside essential spectrum V: on absorbing points of coupling resonances

Let $H_0$ and $V$ be self-adjoint operators, such that $V$ admits a factorisation $V = F^*JF$ with bounded self-adjoint $J$ and $|H_0|^{1/2}$-compact $F.$ Coupling resonance functions, $r_j(z),$ of the pair $H_0$ and $V$ can be defined as $r_j(z) = - σ_j(z) ^{-1},$ where $σ_j(z)$ are eigenvalues of the compact-operator valued holomorphic function $F(H_0-z)^{-1} F^*J.$ Taken together, the functions $r_j(z)$ form an infinite-valued holomorphic function on the resolvent set of~$H_0.$ These functions contain a lot of information about the pair $H_0, V$ (this is well-known in the case of rank one $…

本质谱内的光谱流

Spectral flow inside essential spectrum

谱流是泛函分析和微分几何中的经典概念，人们对它有不同的解释，如Fredholm指标、Witten指标和Maslov指标。经典理论处理本质谱之外的光谱流。在本质谱内，频谱移动函数可以被认为是频谱流的适当模拟，但与频谱流不同，频谱移动函数不是整数值函数。本文证明了谱流的概念对a.e.有一个自然的整数值扩张。对本质谱内的光谱参数值也进行了分析，并提出了相应的理论。本文给出的本质谱内谱流的定义适用于经典谱流，从而又给出了一种新的谱流定义。

Spectral flow inside essential spectrum III: coupling resonances near essential spectrum

Given a self-adjoint operator $H_0$ and a relatively $H_0$-compact self-adjoint operator $V,$ the functions $r_j(z) = - σ_j^{-1}(z),$ where $σ_j(z)$ are eigenvalues of the compact operator $(H_0-z)^{-1}V,$ bear a lot of important information about the pair $H_0$ and $V.$ We call them coupling resonances. In case of rank one (and positive) perturbation $V,$ there is only one coupling resonance function, which is a Herglotz function. This case has been studied in depth in the literature, and appears in different situations, such as Sturm-Liouville theory, random Schrödinger operators, harnomic a…

利用Neumann-Poincaré算子的谱性质分析光滑区域上的等离子体共振

Analysis of plasmon resonance on smooth domains using spectral properties of the Neumann-Poincaré operators

定量研究了光滑区域上Neumann-Poincaré算子在本征值处的等离子体共振和本征值的聚点本质谱。我们首先推广对称化原理，使单层势成为一个酉算子从$H^{-1/2}$到$H^{1/2}$。然后，我们表明，在本质谱共振弱于本征值。结果表明，异常局域共振发生在椭圆上的本质谱，但不像在[20]中考虑的核壳结构，在椭圆上发生掩盖。结果表明，在三维球上，在本质谱处不发生共振。

Spectral flow inside essential spectrum IV: $F^*F$ is a regular direction

Let~$H_0$ and~$V$ be self-adjoint operators such that~$V$ admits a factorisation $V = F^*JF$ with bounded self-adjoint $J$ and $|H_0|^{1/2}$-compact~$F.$ Flow of singular spectrum of the path of self-adjoint operators $H_0 + rV,$ $r \in \mathbb R,$ -- also called spectral flow, through a point $λ$ outside the essential spectrum of~$H_0$ is well studied, and appears in such diverse areas as differential geometry and condensed matter physics. Inside the essential spectrum the spectral flow through $λ$ for such a path is well-defined if the norm limit $$ \lim_{y \to 0^+} F (H_0 + r V - λ- iy)^{-1…

涂层结构线弹性反常局部共振隐身

Cloaking by anomalous localized resonance for linear elasticity on a coated structure

我们研究了弹性静力系统中圆形涂层结构和与之相关的隐身上的异常局部共振。该结构由具有恒定Lamé参数的圆形核和与核的Lamé参数成比例的负Lamé参数的圆形壳组成。我们证明了对应于该结构的Neumann-Poincaré算子的特征值收敛到由Lamé参数确定的某些非零数，并推导出收敛的精确渐近性。然后，我们表明与估计，伪装异常局部共振发生时，只有当偶极型源位于临界半径内的核和壳的半径确定。