Root Vectors for Geometrically Simple Multiparameter Eigenvalues

A class of multiparameter eigenvalue problems involving (generally) non self-adjoint and unbounded operators is studied. Bases for lower order root subspaces, at geometrically simple eigenvalues of Fredholm type of arbitrary finite index, are computed in terms of the underlying multiparameter system.     

Eigenvalue Problems for Fredholm Integral Operators

The calculation of an approximation to the eigenelements ofa Fredholm integral operator can be reduced, using the Nyströmmethod, to the solution of an algebraic eigenvalue problem.To reduce the size of the system of equations to be solved,we propose two iterative variants. In the case of a self-adjointoperator, a special integration rule can be used to regularizethe system of equations. In this case, orthogonality relationscan be introduced for close or multiple eigenvalues.     

Quadratic Eigenvalue Problems for Second Order Systems

We consider the spectral structure of a quadratic second order system boundary-value problem. In particular we show that all but a finite number of the eigenvalues are real and semi-simple. We develop the eigencurve theory for these problems and show that the order of contact between an eigencurve and the parabola gives the Jordan chain associated with the eigenvector corresponding to that eigencurve. Following this we use our knowledge of the eigencurves to obtain eigenvalue asymptotics. Finally the completeness of the eigenfunctions is studied using operator matrix techniques. It should be noted here that the usual left definiteness assumptions have been overcome in this study.     

Fredholm积分方程特征值问题配置法外推的Matlab实验

通过Matlab编程作数值实验,探讨了Fredholm积分方程特征值问题配置法的Richardson外推规律,指出利用Richardson外推可以进一步提高近似特征值λh的精度阶.     

Using Root Functions for Eigenvalue Problems of Ordinary Differential Operators

We study eigenvalue problems for an ordinary differential operator L acting on L ₂(?)-spaces (Problem 1) and on L ₂(J)-spaces (Problem 2). Here J is a bounded but large interval. Assuming that in Problem 1 the spectral parameter s lies in the set of normal points of L, we show that the structure of eigenspaces for both problems is similar to the structure of finite complex-valued matrices. In the case of a finite matrix, the geometry of eigenspaces is described by the Jordan form. In the case of ordinary differential operators, the corresponding geometry is described by a sequence of root functions. Therefore, the main tool of our studies is root functions for complex-valued analytical matrix functions.     

On Eigenvalue Intervals for Discrete Second Order Boundary Value Problems

In this paper, we consider discrete second order three-point boundary value problems. By exploring the properties of the associated Green's function and applying Guo-Krasnosel'skii's fixed point theorem, we show the existence of eigenvalue intervals.     

Two-Parameter Eigenvalue Problems in Nonlinear Second Order Differential Equations

Two-parameter nonlinear second order differential equations are studied. By using a variational method we characterize the variational eigenvalues μ = μ(λ) and study the properties of μ(λ). Furthermore, asymptotic formulas of μ(λ) as λ → ±∞ are established.     

解二阶椭圆本征值问题的有限元插值校正方法

§1.引言 本文介绍解二阶椭圆本征值问题的有限元插值校正方法.理论分析和数值实验都表明,此方法具有低代价和高精度的特点.这个方法是超收敛和迭代伽略金法的思想相结合的产物。 考虑二阶椭圆本征值问题:     

Multiparameter perturbation theory of Fredholm operators applied to bloch functions

In the present paper, a family of linear Fredholm operators depending on several parameters is considered. We implement a general approach, which allows us to reduce the problem of finding the set Λ of parameters t = (t ₁, ..., t _n ) for which the equation A(t)u = 0 has a nonzero solution to a finite-dimensional case. This allows us to obtain perturbation theory formulas for simple and conic points of the set Λ by using the ordinary implicit function theorems. These formulas are applied to the existence problem for the conic points of the eigenvalue set E(k) in the space of Bloch functions of the two-dimensional Schrödinger operator with a periodic potential with respect to a hexagonal lattice.     

Landesman-Lazer Type Problems At An Eigenvalue Of Odd Multiplicity

This paper is concerned with sublinear perturbations of resonant linear problems (Landesman-Lazer problems). We establish some a priori bounds and use these together with Leray-Schauder continuation and bifurcation arguments to obtain extensions of some known results where the nonlinear perturbation terms are bounded.     

特征值问题的Lagrange型各向异性有限元方法

在各向异性网格下首先研究了二阶椭圆特征值问题算子谱逼近的若干抽象结果.然后将这些结果具体应用于线性和双线性Lagrange型协调有限元,得到了与传统有限元网格剖分下相同的最优误差估计,从而拓宽了已有的成果.     

Eigenvalue Problems for Hemivariational Inequalities

We consider a semilinear eigenvalue problem with a nonsmooth potential (hemivariational inequality). Using a nonsmooth analog of the local Ambrosetti–Rabinowitz condition (AR-condition), we show that the problem has a nontrivial smooth solution. In the scalar case, we show that we can relax the local AR-condition. Finally, for the resonant λ?=?λ ₁ problem, using the nonsmooth version of the local linking theorem, we show that the problem has at least two nontrivial solutions. Our approach is variational, using minimax methods from the nonsmooth critical point theory.     

Algorithms for Inverse Eigenvalue Problems

Two new algorithms based on QR decompositions (QRDs) (with column pivoting) are proposed for solving inverse eigenvalue problems, and under some non-singularity assumptions they are both locally quadratically convergent. Several numerical tests are presented to illustrate their convergence behavior.     

二阶退化双曲型方程的Darboux型问题

本文讨论二阶退化双曲型方程的一些边值问题.文中先给出第一Darboux问题和一般斜微商边值问题的提法和解的表示式,然后使用复分析方法证明了上述问题解的存在唯一性.     

一类Monge-Ampère方程的特征值问题

对一类Monge-Amp(e)re方程的特征值问题进行了研究.通过移动平面法证明了在凸对称区域内,Dirichlet问题的C2凹(凸)解一定是对称的.进而通过对常微分方程和椭圆形偏微分方程的讨论,得到一类n维单位球上特征值问题的非平凡解的存在性和正则性结果.     

Inverse Eigenvalue Problems for Smooth Potential

We consider the inverse eigenvalue problem of the onedimensional Schrödinger operator for finite intervals. We give sufficient conditions for finitely many partially known spectra and partial information on the potential to determine the Schrodinger operator on the whole interval.     

Finite Volume Methods for Eigenvalue Problems

Based on a linear finite element space, in this paper, two symmetric finite volume schemes are proposed for self-adjoint elliptic boundary eigenvalue problems. Both convergence and superconvergence are discussed.This revised version was published online in October 2005 with corrections to the Cover Date.     

Greedy Algorithms for High-Dimensional Eigenvalue Problems

In this article, we present two new greedy algorithms for the computation of the lowest eigenvalue (and an associated eigenvector) of a high-dimensional eigenvalue problem and prove some convergence results for these algorithms and their orthogonalized versions. The performance of our algorithms is illustrated on numerical test cases (including the computation of the buckling modes of a microstructured plate) and compared with that of another greedy algorithm for eigenvalue problems introduced by Ammar and Chinesta.     

树的第二个最大特征值

本文得到了边独立数为n且阶为2n+2的树的第二个最大特征值的精确上界,且给出了达到上界的所有的极树.     

Lower Bounds for Non-Classical Eigenvalue Problems

Several lower bounds on the ground state energy of Schrödinger operators with non-confining potentials \({V_{\alpha}(x) = {\prod_{j=1}^{n}} |x_{j}|^{\alpha_{j}}}\), \({\alpha_{j} > 0}\), on \({{\mathbb R}^{n}}\) are obtained. These results lead to explicit (computable) strictly positive lower bounds and can be generalized to certain ‘product’-potentials.