带有谱参数边界条件且权函数变号的不连续Sturm-Liouville算子

研究了一类边界条件中含有谱参数且权函数变号的不连续Sturm-Liouville算子L.首先构造了一个与边值问题相关联的Krein空间K和新算子A使得所考虑的算子L与新算子A的特征值相同,证明了新算子A在Krein空间K中是自共轭的.进一步地,通过研究算子A的谱分布,得到了该边值问题有可数个实的特征值、它们是上下无界的...     

带有三个转移条件的Sturm-Liouville有限谱问题及其矩阵表示

主要研究带有三个转移条件的Sturm-Liouville有限谱问题.首先通过构造一类正则的带有三个转移条件的Sturm-Liouville问题,验证其恰有nl个特征值,进而表明带有三个转移条件的Sturm-Liouville问题等价于一类矩阵特征值问题,且其具有相同的特征值.此外,证明了这nl个特征值在非自共轭边界条件下可位于复平面内任何位置,在自共轭边界条件下可位于实轴上任何位置的结论.分析的关键是判断函数的迭代,运用的主要工具是Rouche定理.     

边界条件中带有谱参数的不连续Sturm-Liouville算子北大核心CSCD

本文研究了具有转移条件且边界条件有特征参数的Sturm-Liouville算子T.首先由算子T本身出发研究其特征值问题,得到了λ是该边值问题的特征值的充要条件.借助新空间H和新算子A,通过构造算子A的Green函数,证明了算子T的特征函数扩张成新算子A的特征函数形成H的标准正交基.     

高阶常微分算子特征值的重数关系

本文借助于边条件空间的几何结构,证明了自伴的高阶常微分算子特征值的解析重数等于几何重数,这是对常型Sturm-Liouville问题相关结果的一个推广.     

左定Sturm-Liouville算子的特征值计算(英文)

研究了定义在有限区间[a,b]上的具有分离型和混合型边界条件的左定正则Sturm-Liouville算子的特征值问题.把具有混合型边界条件的左定正则Sturm-Liouville问题转化成二维的、具有分离型边界条件的右定正则Sturm-Liouville问题,给出了具有混合型边界条件的左定正则Sturm-Liouville算子的特征值的数值计算方法.     

时标上具有分布势函数的Sturm-Liouville问题的矩阵表示

本文讨论时标上具有分布势函数的二阶Sturm-Liouville问题的矩阵表示.通过分析得出所研究的具有分布势函数的Sturm-Liouville问题与一类矩阵特征值问题之间的等价关系.文章针对分离型和实耦合型自共轭边界条件分别进行了讨论.     

带有多点边条件的高阶微分算子的自共轭性

研究带有多点边条件的Sturm-Liouville问题,在新的Hilbert空间中,定义与多点边条件中连接特性相关的最大算子和最小算子,建立了带有多点边条件的高阶微分算子自共轭性的解析判别准则.     

自共轭算子束的谱及其应用

首先研究了自共轭算子束L—λV的谱曲线,其中L和V是Hilbert空间H内的自共轭算子.其次研究了谱问题Ly=λVy的特征值.最后,将所得的结论应用到正则和奇异的常微分算子的不定谱问题中.     

边界条件中带有谱参数的不连续Sturm-Liouville算子的特征值问题

本文研究了具有转移条件且边界条件含特征参数的Sturm-Liouville算子L的特征值问题.首先,使用微分算子谱分析经典的方法,得到λ是该边值问题的特征值的充要条件,证明了该边值问题最多有可数个实的特征值、没有有限值的聚点.其次,通过渐近估计证得,所研究的Sturm-Liouville算子L有可数个离散的特征值且下方有界.     

广义Rayleigh商与有限元法2-网格离散方案

研究广义Rayleigh商和高效率有限元计算方案,做了下列工作：1）把Rayleigh商加速技巧推广到非自共轭问题,定义了算子型广义Rayleigh商和弱形式型广义Rayleigh商,并建立了近似特征向量及其广义Rayleigh商之间的基本关系式.2）在误差估计式中用有限元特征值的陡度取代准确特征值的陡度,得到新的误差估计式.3）在许进超和周爱辉工作的基础上建立了解非自共轭椭圆微分算子特征值问题的有限元2-网格离散方案,并用于协调有限元法和非协调有限元法.从理论分析和数值实验两个方面证明了2-网格方案的有效性.4）把解自共轭椭圆微分算子特征值问题的迭代Galerkin法、插值校正法和梯度重构法推广到非自共轭椭圆微分算子特征值问题.     

二部图的单特征值

一个图的特征值通常指的是它的邻接矩阵的特征值,在图的所有特征值中,重数为1的特征值即所谓的单特征值具有特殊的重要性.确定一个图的单特征值是一个比较困难的问题,主要是没有一个通用的方法.1969年,Petersdorf和Sachs给出了点传递图单特征值的取值范围,但是对于具体的点传递图还需要根据图本身的特性来确定它的单特征值.给出一类正则二部图,它们是二面体群的凯莱图,这类图的单特征值中除了它的正、负度数之外还有0或者±1,而它们恰好是Petersdorf和Sachs所给出的单特征值范围内的中间取值.     

Singular left-definite Sturm-Liouville problems

We study singular left-definite Sturm-Liouville problems with an indefinite weight function. The existence of eigenvalues is established based on the existence of eigenvalues of corresponding right-definite problems. Furthermore, for each singular left-definite problem with limit-circle non-oscillatory endpoints we construct a regular left-definite problem with the same eigenvalues and use it to obtain properties of eigenvalues and eigenfunctions. Inequalities among eigenvalues recently established for regular left-definite problems are extended to the singular case.     

A method for separating nearly multiple eigenvalues for Hermitian matrix

In this paper, we propose a numerical method to verify for nearly multiple eigenvalues of a Hermitian matrix not being strictly multiple eigenvalues. From approximate eigenvalues computed, it seems to be difficult to distinguish whether they are strictly multiple eigenvalues or simple ones, and if they are very close each other, the verification method for simple eigenvalues may fail to enclose them separately, because of singularity of the system in the verification. There are several methods for enclosing multiple and nearly multiple eigenvalues (e.g., [Rump, Computational error bounds for multiple or nearly multiple eigenvalues, Linear Algebra Appl. 324 (2001) 209–226]), For such cases, there is no result to decide the enclosed eigenvalues are nearly multiple or strictly multiple, up to now. So, for enclosed eigenvalues, we propose a numerical method to separate nearly multiple eigenvalues.     

Eigenvalues of second-order difference equations with coupled boundary conditions

This paper is concerned with coupled boundary value problems for self-adjoint second-order difference equations. Existence of eigenvalues is proved, numbers of eigenvalues are calculated, and relationships between the eigenvalues of a self-adjoint second-order difference equation with three different coupled boundary conditions are established. These results extend the relevant existing results of periodic and antiperiodic boundary value problems.     

矩阵特征值的一类新的包含域

用盖尔圆盘定理来估计矩阵的特征值是一个经典的方法,这种方法仅利用矩阵的元素来确定特征值的分布区域.本文利用相似矩阵有相同的特征值这一理论,得到了矩阵特征值的一类新的包含域,它们与盖尔圆盘等方法结合起来能提高估计的精确度.     

Error estimate of eigenvalues of discrete linear Hamiltonian systems with small perturbation

In this paper, eigenvalues of perturbed discrete linear Hamiltonian systems are considered. A new variational formula of eigenvalues is first established. Based on it, error estimates of eigenvalues of systems with small perturbation are given under certain non-singularity conditions. Small perturbations of the coefficient functions, the weight function and the coefficients of the boundary condition are all involved. As a direct consequence, continuous dependence of eigenvalues on boundary value problems is obtained under the non-singularity conditions. In addition, two examples are presented to illustrate the necessity of the non-singularity conditions and the complexity of the problem in the singularity case.     

Distribution of eigenvalues and eigenvectors of Wishart matrix when the population eigenvalues are infinitely dispersed and its application to minimax estimation of covariance matrix

We consider the asymptotic joint distribution of the eigenvalues and eigenvectors of Wishart matrix when the population eigenvalues become infinitely dispersed. We show that the normalized sample eigenvalues and the relevant elements of the sample eigenvectors are asymptotically all mutually independently distributed. The limiting distributions of the normalized sample eigenvalues are chi-squared distributions with varying degrees of freedom and the distribution of the relevant elements of the eigenvectors is the standard normal distribution. As an application of this result, we investigate tail minimaxity in the estimation of the population covariance matrix of Wishart distribution with respect to Stein's loss function and the quadratic loss function. Under mild regularity conditions, we show that the behavior of a broad class of tail minimax estimators is identical when the sample eigenvalues become infinitely dispersed.     

Spectral Analysis of an Operator Arising in Fluid Dynamics

Eigenmode solutions are very important in stability analysis of dynamical systems. The set of eigenvalues of a non-self-adjoint differential operator originated from the linearization of some Cauchy problem is investigated. It is shown that the eigenvalues are purely imaginary, and that they are related to the eigenvalues of Heun's differential equation. These two results are used to derive the asymptotic behavior of the eigenvalues and to compute them numerically.     

Numerical enclosure for multiple eigenvalues of an Hermitian matrix whose graph is a tree

In this paper we consider a numerical enclosure method for multiple eigenvalues of an Hermitian matrix whose graph is a tree. If an Hermitian matrix A whose graph is a tree has multiple eigenvalues, it has the property that matrices which are associated with some branches in the undirected graph of A have the same eigenvalues. By using this property and interlacing inequalities for Hermitian matrices, we show an enclosure method for multiple eigenvalues of an Hermitian matrix whose graph is a tree. Since we do not generally know whether a given matrix has exactly a multiple eigenvalue from approximate computations, we use the property of interlacing inequalities to enclose some eigenvalues including multiplicities.In this process, we only use the enclosure of simple eigenvalues to enclose a multiple eigenvalue by using a computer and interval arithmetic.     

一类具有转移条件的向量Sturm-Liouville问题的特征值

考虑了一类具有转移条件的向量Sturm-Liouville问题的特征值及其重数问题.首先构造了与问题相关的新内积和基本解,得到特征值的充要条件.在此基础上证明了二维情况下,问题特征值的代数重数与几何重数相等.