Calculation of eigenvalues of a discrete self-adjoint operator perturbed by a bounded operator

We generalize the method of regularized traces which calculates eigenvalues of a perturbed discrete operator for the case of an arbitrary multiplicity of eigenvalues of the unperturbed operator. We obtain a system of equations, enabling one to calculate eigenvalues of the perturbed operator with large ordinal numbers. As an example, we calculate eigenvalues of a perturbed Laplace operator in a rectangle.     

Asymptotics of the eigenvalues and the formula for the trace of perturbations of the Laplace operator on the sphere \mathbb{S}^2

In this paper, we study the asymptotics of the eigenvalues of the Laplace operator perturbed by an arbitrary bounded operator on the sphere . For the first time, for the partial differential operator of second order, the leading term of the second correction of perturbation theory is obtained. A connection between the coefficient of the second term of the asymptotics of the eigenvalues and the formula for the traces of the operator under consideration is established.Translated from Matematicheskie Zametki, vol. 77, no. 3, 2005, pp. 434–448Original Russian Text Copyright © 2005 by V. A. Sadovnichii, Z. Yu. Fazullin.This revised version was published online in April 2005 with a corrected issue number.     

Universal inequalities for lower order eigenvalues of self-adjoint operators and the poly-Laplacian

In this paper, we first establish an abstract inequality for lower order eigenvalues of a self-adjoint operator on a Hilbert space which generalizes and extends the recent results of Cheng et al. （Calc. Var. Partial Differential Equations, 38, 409-416 （2010））. Then, making use of it, we obtain some universal inequalities for lower order eigenvalues of the biharmonic operator on manifolds admitting some speciM functions. Moreover, we derive a universal inequality for lower order eigenvalues of the poly-Laplacian with any order on the Euclidean space.     

Efimov's Effect in a Model of Perturbation Theory of the Essential Spectrum

A model operator similar to the energy operator of a system with a nonconserved number of particles is studied. The essential spectrum of the operator is described, and under some natural conditions on the parameters it is shown that there are infinitely many eigenvalues lying below the bottom of the essential spectrum.     

Minimal compact global attractor for a damped semilinear wave equation

Abstract

The asymptotic behavior of eigenvalues of an elliptic operator with a divergence form is discussed. The coefficients of the operator are discontinuous through a boundary of a subdomain and degenerate to zero on the subdomain when a parameter tends to zero. We will prove that the eigenvalues approach eigenvalues of the Laplacian on the subdomain or on the complement. We will obtain precise asymptotic behavior of their convergence.     

Eigenvalues of a Differential Operator and Zeros of the Riemann $\zeta$-Function

The eigenvalues of a differential operator on a Hilbert-Pόlya space are determined. It is shown that these eigenvalues are exactly the nontrivial zeros of the Riemann $\zeta$-function. Moreover, their corresponding multiplicities are the same.     

混合气体线性化Boltzmann算子的特征值与特征函数

考虑了混合气体线性化Boltzmann算子的特征值与特征函数问题.通过对算子中核函数的分析得出了更一般的混合气体线性化Boltzmann算子的特征值与特征函数.而且结论还适用于单一气体情况.     

Dirac Eigenvalues for Generic Metrics on Three-Manifolds

We show that for generic Riemannian metrics on a closed spin manifold of dimension three the Dirac operator has only simple eigenvalues.     

ESTIMATES OF EIGENVALUES FOR UNIFORMLY ELLIPTIC OPERATOR OF SECOND ORDER

ＥＳＴＩＭＡＴＥＳＯＦＥＩＧＥＮＶＡＬＵＥＳＦＯＲＵＮＩＦＯＲＭＬＹＥＬＬＩＰＴＩＣＯＰＥＲＡＴＯＲＯＦＳＥＣＯＮＤＯＲＤＥＲＱＩＡＮＣＨＵＮＬＩＮ（钱椿林）ＣＨＥＮＺＵＣＨＩ（陈祖墀）（ＤｅｐａｒｔｍｅｎｔｏｆＭａｔｈｅｔｎａｔｉｃｓ，Ｕｎｉｖｅｒ...     

带势加权散度形式的 Grushin 型退化椭圆算子的 Dirichlet 特征值的上界*

本文考虑了带有位势的散度形式的 Grushin 型退化椭圆算子的 Dirichlet 加权特征值的估计.利 用傅里叶变换的方法得到了特征值的精确下界估计.然后通过试验函数的方法得到了特征值上界的杨型 不等式.     

On the spectral properties of an operator pencil

For a broad class of iterative algorithms for solving saddle point problems, the study of the convergence and of the optimal properties can be reduced to an analysis of the eigenvalues of operator pencils of a special form. A new approach to analyzing spectral properties of pencils of this kind is proposed that makes it possible to obtain sharp estimates for the convergence rate.     

一类四阶Hamilton算子特征值的代数指标

研究了一类四阶Hamilton算子H_A特征值的代数指标问题.根据算子A与Hamilton算子H_A的关系,讨论了Hamilton算子H_A特征值的几何重数,代数指标及代数重数.最后结合例子说明其结果的有效性.     

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.     

On the eigenvalues for a weighted p-Laplacian operator on metric graphs

We provide an explicit upper bound of the eigenvalues corresponding to a weighted p-Laplacian operator defined on a connected metric graph with finite total length.     

一类积—微分算子的离散本征值

本文讨论的积-微分算子是一类以众多应用领域为背景的无界、非自伴线性算子.在较一般的假设下,籍助L~2空间的线性算子理论,我们证明了这类算子存在离散本征值的充分条件,并获得了可供实际工作者参考的估计式.     

一类具有转移条件的不连续微分算子的渐近状态

研究一类具有转移条件且边界条件依赖于特征参数的Sturm-Liouville算子,建立一个与其相关的新的空间框架,给出其特征的相关性质与特征函数的完备性,利用函数论方法得出其特征的渐近表示,并获得Green函数的表达式.     

一类具抽象边界条件的迁移算子的谱分布

利用线性算子半群理论,研究了板几何中具抽象边界条件的各向异性、连续能量、非均匀介质的迁移方程．在假设边界算子日部分光滑和扰动算子K正则的条件下,采用豫解方法,得到了该迁移算子A的谱在区域Г中由至多可数个具有限代数重数的离散本征值组成等结果．     

Algebraic Properties of Subdivision Operators with Matrix Mask and Their Applications

Subdivision operators play an important role in wavelet analysis. This paper studies the algebraic properties of subdivision operators with matrix mask, especially their action on polynomial sequences and on some of their invariant subspaces. As an application, we characterize, under a mild condition, the approximation order provided by refinable vectors in terms of the eigenvalues and eigenvectors of polynomial sequences of the associated subdivision operator. Moreover, some necessary conditions, in terms of nondegeneracy and simplicity of eigenvalues of a matrix related to the subdivision operator for the refinable vector to be smooth are given. The main results are new even in the scalar case     

Local Perturbations of the Schrödinger Operator on the Axis

We present necessary and sufficient conditions for the existence of eigenvalues of the Schrödinger operator on an axis under small local perturbations. In the case where the eigenvalues exist, we construct their asymptotic approximations.     

On absence of embedded eigenvalues for schrÖdinger operators with perturbed periodic potentials

The problem of absence of eigenvalues imbedded into the continuous spectrum is considered for a Schrödinger operator with a periodic potential perturbed by a sufficiently fast decaying "impurity" potential. Results of this type have previously been known for the one-dimensional case only. Absence of embedded eigenvalues is shown in dimensions two and three if the corresponding Fermi surface is irreducible modulo natural symmetries. It is conjectured that all periodic potentials satisfy this condition. Separable periodic potentials satisfy it, and hence in dimensions two and three Schrödinger operator with a separable periodic potential perturbed by a sufficiently fast decaying "impurity" potential has no embedded eigenvalues