Albrecht Seelmann, Matthias Täufer, Krešimir Veselić
We classify all sets of the form $\bigcup_{t\in\mathbb{R}}\mathrm{spec}(A+tB)$ where $A$ and $B$ are self-adjoint operators and $B$ is bounded, non-negative, and non-zero. We show that these sets are exactly the complements of discrete subsets of $\mathbb{R}$, that is, of at most countable subsets of $\mathbb{R}$ that contain none of their accumulation points.
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A canonical factorization is given for a quadratic pencil of accretive operators in a Hilbert space. Also, we establish some relationships between an m-accretive operator and its Moore-Penorse inverse. As an application, we study a result of existence, uniqueness, and maximal regularity of the strict solution for complete abstract second order differential equation in the non-homogeneous case. The paper is concluded with some questions left open from the preceding discussions.
This paper is concerned with the reduction of the spectral problem for symmetric linear operator pencils to a spectral problem for the single operator. Also, a Rayleigh-Ritz-like bounds on eigenvalues of linear operator pencils are obtained.
We introduce concepts of essential numerical range for the linear operator pencil $λ\mapsto A-λB$. In contrast to the operator essential numerical range, the pencil essential numerical ranges are, in general, neither convex nor even connected. The new concepts allow us to describe the set of spectral pollution when approximating the operator pencil by projection and truncation methods. Moreover, by transforming the operator eigenvalue problem $Tx=λx$ into the pencil problem $BTx=λBx$ for suitable choices of $B$, we can obtain non-convex spectral enclosures for $T$ and, in the study of truncati…
We establish some relationships between an m-accretive operator and its Moore-Penorse inverse. We derive some perturbation result of the Moore-Penorse inverse of a maximal accretive operator. As an application we give a factorization theorem for a quadratic pencil of accretive operators. Also, we study a result of existence, uniqueness, and maximal regularity of the strict solution for complete abstract second order differential equation. Illustrative examples are also given.
An inverse problem for the quadratic pencil of non-self-adjoint matrix operators on the half-line
考虑了射线上的非自伴随矩阵Sturm-Liouville算子束，研究了由其Weyl矩阵构造这束算子束的逆问题.证明了一个唯一性定理，并给出了求解的构造性算法。
Pseudospectra of the heat operator Pencil
该研究针对带左狄利克雷、右诺依曼边界条件的一维热方程，将其重构为非自伴2×2无界块算子矩阵束，通过分析其谱、伪谱及(n,ε)-伪谱包络研究热算子束，并辅以离散化方程图像验证结果。
The relation between the spectra of operator pencils with unbounded coefficients and of associated linear relations is investigated. It turns out that various types of spectrum coincide and the same is true for the Weyr characteristics. This characteristic describes how many independent Jordan chains up to a certain length exist. Furthermore, the change of this characteristic subject to one-dimensional perturbations is investigated.
Hermitian Pencils and their Representation in Krein Spaces
研究形如$\mathcal{A}({\lambda}) = {\lambda}E - A$的铅笔谱性质，借助克莱因空间中线性关系，通过值域表示，在对$A$和$E$的假设下，利用线性关系谱性质推导算子铅笔谱性质。
Spectral theory of infinite dimensional dissipative Hamiltonian systems
本文研究无限维耗散哈密顿系统中算子束的谱理论，引入三种奇异性概念，分析其与微分代数方程解唯一性的关系，并揭示在耗散结构化算子束中唯一性与奇异性紧密相关。
Spectral and oscillation properties for a linear pencil of fourth-order differential operators
本文研究线性束$A-λB$的谱和振荡性质。这里，$A$和$B$是分别由微分表达式$(py")"$和$-y"+ cry$生成的线性算子。特别地，它表明，该问题的负特征值是简单的和相应的特征函数$y_{-n}$有$n-1$零在$(0,1)$。
Quadratic operator pencils associated with the conservative Camassa-Holm flow
我们讨论了Sturm-Liouville型问题的正谱和逆谱理论，该问题对特征值参数具有二次依赖性，这是保守Camassa-Holm流的等谱问题。
Existence of eigenvalues of a linear operator pencil in a curved waveguide -- localized shelf waves on a curved coast
研究弯曲海岸上存在非传播的、被捕获的大陆架波（CSW）的可能性，在数学上归结为一个弯曲带中自伴算子束的谱问题。使用的方法中开发的设置的波导捕获模式的问题，我们表明，这样的CSWs存在广泛的一类海岸曲率和深度剖面。
Schrodinger operators and associated hyperbolic pencils
对于一大类Schrodinger算子，我们通过使耦合常数以非常特殊的方式依赖于能量，引入了双曲二次束。对于这些铅笔，散射理论的许多问题都更容易研究。然后，我们给出了一些应用到原来的薛定谔算子。
The characterization problem for one class of second order operator pencil with complex periodic coefficients
本文的目的是解决一类复周期系数二阶算子束的特征化问题，包括确定散射数据上保证重构势属于某一类的充要条件.
Spectral analysis for one class of second-order indefinite non-self-adjoint differential operator pencil
研究了轴上具有间断系数的复周期位势微分算子束的反问题。研究了该算子的基本解的主要特征，研究了该算子的谱.我们给出了反问题的形式，证明了一个唯一性定理，并给出了反问题解的构造性步骤
We identify a class of operator pencils, arising in a number of applications, which have only real eigenvalues. In the one-dimensional case we prove a novel version of the Sturm oscillation theorem: if the dependence on the eigenvalue parameter is of degree $k$ then the real axis can be partitioned into a union of $k$ disjoint intervals, each of which enjoys a Sturm oscillation theorem: on each interval there is an increasing sequence of eigenvalues that are indexed by the number of roots of the associated eigenfunction. One consequence of this is that it guarantees that the spectrum of these …
This article can be considered as the first version of a book which the author plans to write about half-range problems in operator theory. It consists of two parts. The first part is based on lectures which the author delivered at University of Calgary and Lomonosov Moscow State University. The main attention in this part is paid to the selection of waves which are involved in the formulation of the Mandelstamm radiation principle (the eigen-pairs, corresponding to the real eigenvalues) and to the factorization problems of self-adjoint and dissipative, quadratic and polynomial operator pencil…
We consider a second order functional-differential pencil with two constant delays of the argument and study the inverse problem of recovering its coefficients from the spectra of two boundary value problems with one common boundary condition. The uniqueness theorem is proved and a constructive procedure for solving this inverse problem along with necessary and sufficient conditions for its solvability is obtained. Moreover, we give a survey on the contemporary state of the inverse spectral theory for operators with delay. The pencil under consideration generalizes Sturm-Liouville-type operato…
Eigenvalues of a one-dimensional Dirac operator pencil
研究了一维Dirac算子束的谱，其中势前的耦合常数被认为是谱参数。受最近石墨烯波导研究的启发，我们重点研究了Dirac算子的核包含平方可积〔的〕的耦合常数的值。在物理学文献中，这样的函数被称为受限零模。得到了引起零模的耦合常数的渐近分布的几个结果。特别是，我们表明，这种分布取决于在一个微妙的方式上的符号变化和存在的差距的潜力。令人惊讶的是，它还取决于由势确定的某些量的算术性质。我们进一步观察到，可变符号的潜力可能会产生复杂的本征值的运营商束。给出了一些例子和数值计算来说明这些现象。
Index Theorems for Polynomial Pencils
本文综述了线性化哈密顿系统特征值和多项式算子束特征值的指数定理。我们提出了一个简单的共同的图形解释和推广的指数理论使用的概念，图形克莱因签名。进一步证明了满足L（λ）u（λ）= μ（λ）u（λ）的算子束L（λ）的特征向量u= u（λ）在L（λ）的特征值处的导数不仅生成L（λ）的根向量的任意链，而且生成携带额外信息的链.
Non-self-adjoint second-order differential pencils on a finite interval with non-separated quasi-periodic boundary conditions and jump conditions are studied. We establish properties of spectral characteristics and investigate the inverse spectral problem of recovering the operator from its spectral data. For this inverse problem we prove the corresponding uniqueness theorem and provide an algorithm for constructing its solution.
We consider a quadratic operator pencil with a small periodic perturbation multiplied by the spectral parameter. It is motivated, in particular, by a one-dimensional Klein-Gordon equation with a time-parity-symmetric perturbation. We study in details the structure of the considered operator pencil. We show that its essential spectrum has a band structure and at certain thresholds, the bands bifurcate into small parabolas. We then study how the isolated limiting eigenvalues behave under the perturbation. We show that if zero is a limiting isolated eigenvalue, under the perturbation it remains a…
Quadratic pencil of difference equations: Jost solutions, spectrum, and principal vectors
本文利用Schr dinger型差分算子$L_λ$的二次束，对二阶非自伴差分方程的谱分析进行了研究.通过引入Jost型解，分析了算子$L_λ$的谱的结构和数量性质，从而得到了Degasperis（\emph{J.Math.Phys} . 11：551- 567，1970）和Bairamov et.等人，（\emph{Quaest. Math.} 26：15- 30，2003）开发的。此外，几个类比之间的差异和$q$ -差异的情况下建立。最后，引入了$L_λ$的主向量，为谱展开奠定了基础。数学学科分类（2000年）：39 A10、39 A12、39 A13
On singular pencils with commuting coefficients
本文研究矩阵多项式谱与系数Taylor谱的关系，证明奇异多项式等价于零点在系数Taylor谱中，但该等价在无穷维算子情形失效，并给出新谱描述及Kronecker形式。
Algebraic Spectral Curves of Two-Channel Operator Pencils with Power-Law Response
本文提出了一类J-self-adjoint因果算子笔，其谱行列式精确编码L-函数的局部欧拉因子，针对g=0和g=1的情况分别证明了Riemann zeta函数和椭圆曲线的欧拉因子编码，并解决了二次扭和惰性素数的算术障碍，最后展示了Hilbert-Pólya实现对于zeta(s)的严格收敛假设。
Eigenwaves in Waveguides with Dielectric Inclusions: Spectrum
我们考虑的基本问题的数学理论的波传播的波导与夹杂物。分析是在一个边界特征值问题的麦克斯韦方程组，这是减少到一个本征值问题的运营商束。我们制定的定义本征波和相关的波使用系统的本征向量和相关的向量的铅笔和证明的频谱正常波形成一个非空的孤立点的一个带中的本地化的最多200个实点。
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