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.     

带权流形上的纽曼特征值估计

讨论一类光滑紧致带权黎曼流形上的纽曼特征值估计问题,假定这类流形具有光滑边界,边界是凸的,而且流形上的Bakery-Emery Ricci曲率具有正的下界.利用了极大模原理去证明热方程解的梯度估计,然后得到热核上界估计.再利用热核与特征值的关系,得到了特征值的下界估计.     

典型几何上Ricci流下的特征值

研究了典型几何上规范Ricci流下Laplace-Beltrami算子第一特征值的发展行为.在每一个Bianchi类中,我们估计了特征值的导数.构造了Ricci流下的单调量并得到了特征值的上下界估计.     

关于Camassa-Holm方程动量密度的紧支集在半轴上的估计

主要考虑在半轴上Camassa-Holm方程解的动量密度紧支集大小的估计,方法是根据区间长度与区间特征值的关系,通过估计第一Dirichlet特征值来估计动量密度紧支集的长度.因为知道动量密度紧支集外解的性态,所以通过估计动量密度支集的大小可以得到方程解的更多信息.     

特征值问题混合有限元法的一个误差估计

设（λh,σh,uh）是一个混合有限元特征对.Babuska和Osborn建立了（λh,uh）的误差估计.本文导出了σh的抽象误差估计式.并把该估计式应用于二阶椭圆特征值问题Raviart-Thomas混合有限元格式和重调和算子特征值问题Ciarlet-Raviart混合有限元格式,得到了一些新的误差估计.     

Laplacian第一特征值整体曲率Pinching的一个结果

紧致流形上Laplacian的第一特征值的下界估计一直以来是人们非常感兴趣的问题之一.本文在整体曲率Pinching较小的条件之下考虑这个问题,得到了相应几何条件之下的Laplacian第一特征值的一个下界估计.     

加权Laplace在积分Ricci曲率条件下的特征值估计

本文研究了积分Ricci曲率条件下加权Laplace算子的第一特征值估计的问题.利用Bochner公式与加权Reilly公式等处理特征值问题的方法,获得了加权Laplace在积分Ricci曲率条件下第一特征值估计下界的估计.     

线性离散事件动态系统的辨识

本文讨论利用输出数据来估计或确定系统矩阵特征值和特征向量问题.首先我们给出了特征值的一个估计,然后证明在一定条件下可以确定系统矩阵的特征值和特征向量,或用极限来表征它们,最后指出了所得到的结果在离散事件动态系统分析和控制中的意义.     

扩散过程依全变差范数的收敛速度

本文使用耦合方法,通过对耦合时间的矩的估计得到紧流形上扩散过程依全变差范数指数式收敛的结果;并利用非零第一特征值与特征函数,给出了另外两个估计.     

一类Dirac方程特征值问题的迹恒等式

讨论了带有非局部边界条件的一维Dirac方程BdY/dx+P(x)Y=λY的特征值问题,其中首先建立了问题的特征值集合与一个整函数u(λ)零点集合的对应,并对Dirac算子的特征值进行了估计,然后借助于一个积分恒等式,采用留数方法,得到了该问题的特征值的迹恒等式.     

Cluster robust error estimates for the Rayleigh-Ritz approximation II: Estimates for eigenvalues

This is the second part of a paper that deals with error estimates for the Rayleigh-Ritz approximations of the spectrum and invariant subspaces of a bounded Hermitian operator in a Hilbert or Euclidean space. This part addresses the approximation of eigenvalues. Two kinds of estimates are considered: (i) estimates for the eigenvalue errors via the best approximation errors for the corresponding invariant subspaces, and (ii) estimates for the same via the corresponding residuals. Estimates of these two kinds are needed for, respectively, the a priori and a posteriory error analysis of numerical methods for computing eigenvalues. The paper’s major concern is to ensure that the estimates in question are accurate and ‘cluster robust’, i.e. are not adversely affected by the presence of clustered, i.e. closely situated eigenvalues among those of interest. The paper’s main new results introduce estimates for clustered eigenvalues whereby not only the distances between eigenvalues in the cluster are not present but also the distances between the cluster and the rest of the spectrum appear in asymptotically insignificant terms only.     

Computing the matrix sign and absolute value functions

We present two algorithms for the computation of the matrix sign and absolute value functions. Both algorithms avoid a complete diagonalisation of the matrix, but they however require some informations regarding the eigenvalues location. The first algorithm consists in a sequence of polynomial iterations based on appropriate estimates of the eigenvalues, and converging to the matrix sign if all the eigenvalues are real. Convergence is obtained within a finite number of steps when the eigenvalues are exactly known. Nevertheless, we present a second approach for the computation of the matrix sign and absolute value functions, when the eigenvalues are exactly known. This approach is based on the resolution of an interpolation problem, can handle the case of complex eigenvalues and appears to be faster than the iterative approach. To cite this article: M. Ndjinga, C. R. Acad. Sci. Paris, Ser. I 346 (2008).     

Heisenberg群上Laplace算子的离散谱

本文证明了Heisenberg群上Laplace算子的Dirichlet特征值的存在性，给出了特征值的估计     

Spectral properties of an even-order differential operator

We present the spectral properties of an even-order differential operator whose domain is described by periodic and antiperiodic boundary conditions or the Dirichlet conditions. We derive an asymptotic formula for the eigenvalues, estimates for the deviations of spectral projections, and estimates for the equiconvergence rate of spectral decompositions. Our asymptotic formulas for eigenvalues refine well-known ones.     

Eigenvalues and Entropy Moduli of Operators in Interpolation Spaces

The paper approaches in an abstract way the spectral theory of operators in abstract interpolation spaces. We introduce entropy numbers and spectral moduli of operators, and prove a relationship between them and eigenvalues of operators. We also investigate interpolation variants of the moduli, and offer a contribution to the theory of eigenvalues of operators. Specifically, we prove an interpolation version of the celebrated Carl–Triebel eigenvalue inequality. Based on these results we are able to prove interpolation estimates for single eigenvalues as well as for geometric means of absolute values of the first n eigenvalues of operators. In particular, some of these estimates may be regarded as generalizations of the classical spectral radius formula. We give applications of our results to the study of interpolation estimates of entropy numbers as well as of the essential spectral radius of operators in interpolation spaces.     

Cluster robust error estimates for the Rayleigh-Ritz approximation I: Estimates for invariant subspaces

This is the first part of a paper that deals with error estimates for the Rayleigh-Ritz approximations to the spectrum and invariant subspaces of a bounded Hermitian operator in a Hilbert or Euclidean space. This part addresses estimates for the angles between the invariant subspaces and their approximations via the corresponding best approximation errors and residuals and, for invariant subspaces corresponding to parts of the discrete spectrum, via eigenvalue errors. The paper’s major concern is to ensure that the estimates in question are accurate and ‘cluster robust’, i.e. are not adversely affected by the presence of clustered, i.e. closely situated eigenvalues in the spectrum. Available estimates of such kind are reviewed and new estimates are derived. The paper’s main new results introduce estimates for invariant subspaces in which the operator may have clustered eigenvalues whereby not only the distances between eigenvalues in the cluster are not present but also the distances between the cluster and the rest of the spectrum appear in asymptotically insignificant terms only.     

Inequalities of Eigenvalues for the Dirac Operator on Compact Complex Spin Submanifolds in Complex Projective Spaces

For a compact complex spin manifold M with a holomorphic isometric embed- ding into the complex projective space,the authors obtain the extrinsic estimates from above and below for eigenvalues of the Dirac operator,which depend on the data of an isometric embedding of M.Further,from the inequalities of eigenvalues,the gaps of the eigenvalues and the ratio of the eigenvalues are obtained.     

Commutators, Spectral Trace Identities, and Universal Estimates for Eigenvalues

Using simple commutator relations, we obtain several trace identities involving eigenvalues and eigenfunctions of an abstract self-adjoint operator acting in a Hilbert space. Applications involve abstract universal estimates for the eigenvalue gaps. As particular examples, we present simple proofs of the classical universal estimates for eigenvalues of the Dirichlet Laplacian, as well as of some known and new results for other differential operators and systems. We also suggest an extension of the methods to the case of non-self-adjoint operators.     

A One-Dimensional Schrödinger Operator with Square-Integrable Potential

We study the spectral properties of a one-dimensional Schrödinger operator with squareintegrable potential whose domain is defined by the Dirichlet boundary conditions. The main results are concerned with the asymptotics of the eigenvalues, the asymptotic behavior of the operator semigroup generated by the negative of the differential operator under consideration. Moreover, we derive deviation estimates for the spectral projections and estimates for the equiconvergence of the spectral decompositions. Our asymptotic formulas for eigenvalues refine the well-known ones.     

Estimates for the Eigenvalues of Matrices

For matrices whose eigenvalues are real (such as Hermitian or real symmetric matrices), we derive simple explicit estimates for the maximal (λ_max) and the minimal (λ_min) eigenvalues in terms of determinants of order less than 3. For 3 × 3 matrices, we derive sharper estimates, which use det A but do not require to solve cubic equations.