具转移条件奇型Sturm-Liouville算子相关的Weyl函数

研究了一端奇异且在内部具有转移条件的Sturm-Liouville算子的Weyl函数,我们给出了相应的Weyl函数的定义,并对Weyl函数性质进行了讨论.     

一类Sturm-Liouville问题特征的渐近分析

考虑[0,1]上带第三类边界条件的S-L问题特征值的渐近表示.利用已有的渐近性结果及Fr啨chet导数技术,对特征值进行了精细的分析,清楚地给出了边界条件中的常数(h,H)对特征值的影响.本文的工作对S-L问题的一类反谱问题及相关微分方程反问题的唯一性结果有着重要的应用,也为专著[4,6]中的某些关键结果提供了一个简化的证明途径.     

奇异非线性Sturm-Liouville问题的正解

本文研究奇异非线性Sturm-Liouville问题其中(Lφ)(x)=(p(x)φ′(x))′+g(x)φ(x),并且允许h(x)在x=0和x=1奇异。应用锥理论和不动点指数方法,在与相应的线性算子第一特征值有关的条件下获得了正解的存在性结果,本质地推广和改进了文献[1-9]中的主要结论。     

Sturm-Liouville问题的m-函数与谱

本文先给出了Sturm-Liouville问题的Weyl-Titchmarsh m-函数对谱参数和端点的导数,进而推出其对于谱参数的分段单调性;然后利用m-函数定义了Sturm-Liouville问题的广义特征多项式,并由此得到Sturm-Liouville问题的谱对边界和参数的导数的一个简短证明.     

非局部Sturm-Liouville问题的特征值

本文研究含有非局部势的非局部Sturm-Liouville特征值问题的谱理论.利用修正的Green函数给出了一般情况下特征值的判别函数、下界估计、几何重数和分布位置.     

Lagrange函数的的两个应用

介绍Lagrange函数在大学数学教学中的两个重要应用,利用Lagrange函数中对偶问题来引入特征值和特征向量的定义,以及在规划中由已知原问题求对偶问题.     

满足H(m)条件的奇异积分算子的交换子的有界性

主要讨论了满足H(m)条件的奇异积分算子与Lipschitz函数的交换子在L~p和Hardy空间的有界性,并把这个结果应用于与薛定谔算子相关的Riesz变换.     

单位圆内亚纯函数的Nevanlinna不等式和相关的奇异半径

应用Ahlfors覆盖曲面的方法,得到几个单位圆内亚纯函数的Nevanlinna基本不等式,应用它们证明了亚纯函数在单位圆内关于小函数的奇异半径的存在性,推广了复平面上亚纯函数奇异方向的相关结论.     

两区间Sturm-Liouville问题Green函数及修正Parseval等式（英文）

本文研究具有转移条件的两区间Sturm-Liouville问题.首先得到参数λ是两区间Sturm-Liouville问题的特征值的充要条件,并证明两区间Sturm-Liouville问题的Green函数按特征函数的展开,最后利用Green函数按特征函数的展开证明了修正Parseval等式.     

关于Bzilevich函数族的一个扩展及其Fekete-Szeg■问题

引进扩展的Bzilevich函数族A(α,β),并讨论该族的Fekete-Szeg(o)不等式,得到准确结果.     

Left-definite spaces of singular Sturm-Liouville problems

The left-definite Hilbert spaces for singular Sturm-Liouville problems of the limit-circle type are explicitly constructed. The construction only uses an explicit form of the left-definite boundary conditions, together with a principle solution and a non-principle solution of the differential equation involved.     

Regular approximations of singular Sturm-Liouville problems

Given any self-adjoint realization S of a singular Sturm-Liouville (S-L) problem, it is possible to construct a sequence {S_r{ of regular S-L problems with the properties (i) every point of the spectrum of S is the limit of a sequence of eigenvalues from the spectrum of the individual members of {S_r{ (ii) in the case when 5 is regular or limit-circle at each endpoint, a convergent sequence of eigenvalues from the individual members of {S_r{ has to converge to an eigenvalue of S (iii) in the general case when S is bounded below, property (ii) holds for all eigenvalues below the essential spectrum of S.     

Oscillation theorems for Sturm-Liouville problems with distribution potentials

The Sturm-Liouville problem

$\begin{array}{*{20}c} { - y'' + q(x)y = \lambda y,} \\ {y(0) = y(1) = 0} \\ \end{array} $

is considered with a singular potential q(x) representing the derivative of a real function from the space L ₂[0, 1] in the distributional sense. Two approaches are developed for the study of oscillation properties of eigenfunctions of this problem. The first approach is based on generalization of methods of the Sturm theory. The second one is based on development of variational principles.

     

Absolute and relative Weyl theorems for generalized eigenvalue problems

Weyl-type eigenvalue perturbation theories are derived for Hermitian definite pencils A-λB, in which B is positive definite. The results provide a one-to-one correspondence between the original and perturbed eigenvalues, and give a uniform perturbation bound. We give both absolute and relative perturbation results, defined in the standard Euclidean metric instead of the chordal metric that is often used.     

Nontrivial solutions of singular superlinear Sturm-Liouville problems

The singular superlinear Sturm-Liouville problems

     

Nontrivial solutions of singular sublinear Sturm-Liouville problems

The singular sublinear Sturm-Liouville problems

     

Spectral exactness and spectral inclusion for singular left-definite Sturm-Liouville problems

We study the approximation of the discrete and continuous spectrum of singular left-definite Sturm-Liouville problems with eigenvalues of regular problems on truncated intervals.     

Automatic solution of regular and singular vector Sturm-Liouville problems

This paper describes the algorithms and theory behind a new code for vector Sturm-Liouville problems. A new spectral function is defined for vector Sturm-Liouville problems; this is an integer valued function of the eigenparameter which has discontinuities precisely at the eigenvalues. We describe numerical algorithms which may be used to compute the new spectral function, and its use as amiss-distance function in a new code which solves automatically a large class of regular and singular vector Sturm-Liouville problems. Vector Sturm-Liouville problems arise naturally in quantum mechanical applications. Usually they are singular. The advantages of the author's code lie in its ability to solve singular problems automatically, and in the fact that the user may specify the required eigenvalue by its index.     

Spectral properties and oscillation theorems for mixed boundary-value problems of Sturm-Liouville type

This paper presents analogs, for certain mixed boundary-value problems, of the spectral and oscillatory properties exhibited by classical Sturm-Liouville systems. These mixed boundary-value problems have Green's functions which are sign consistent for all even and/or odd orders.     

The computation of negative eigenvalues of singular Sturm-Liouville problems

In this work we shall develop a new interpolation method forthe computation of eigenvalues of singular Sturm–Liouvilleproblems. Basic properties of the Jost solutions are used todetermine the growth of the boundary function and an appropriateinterpolation basis. This leads to a good approximation of thenegative eigenvalues.