具有混合型边界条件的左定Sturm-Liouville问题特征值的下标计算(英文)

本文研究具有混合型边界条件的左定Sturm-Liouvile问题特征值的下标计算问题.首先给出具有分离型边界条件和混合型边界条件的左定Sturm-Liouville问题的特征值之间的不等式;然后利用这个结果给出一种计算混合型边界条件下左定Sturm-Liouville问题特征值下标的方法.     

一类四阶左定Sturm-Liouville算子特征值的计算问题

主要研究一类四阶左定Sturm-Liouville问题特征值的计算问题.主要方法是将具有混合型边界条件的一般四阶左定正则自伴问题转化成向量型具有分离型边界条件的四阶右定正则自伴问题,这为具有混合型边界条件的一般四阶左定正则问题的特征值的数值计算提供了理论依据.     

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.     

左定四阶微分算子的特征值计算

讨论了一类具有耦合边界条件的左定四阶微分算子,利用具有耦合边界条件的左定四阶微分算子和其相应的右定四阶微分算子的关系,最终给出左定四阶微分算子特征值的计算方法.     

一类奇型Sturm-Liouville算子的逆问题

研究了奇型Sturm-Liouville算子的逆问题.对于固定的n∈N,证明了Sturm-Liouville问题(1.3)-(1.5)的第n个特征值λ_n(q,H)关于H是严格单调增加的,及一组不同边界条件下的第n个特征值的谱集合{λ_n(q,H_k)}_(k=1)~(+∞)能够唯一确定(0,πr)上的势函数q(x).     

求解一类不连续的Sturm-Liouville问题

研究一类边界条件中有谱参数的不连续的Sturm-Liouville问题.首先在Hilbert空间中定义了一个自共轭的线性算子A,使得该类Sturm-Liouville问题的特征值与算子A的特征值相一致.进一步证明了算子A是自共轭的,且这类Sturm-Liouville问题特征值是解析单的.最后展示了一个具体问题的特征值以及特征函数的逼近解.     

左定与定型Sturm-Liouville问题间特征值不等式

应用左定和定型Sturm-Liouville问题特征值的Prüfer角刻画,以及其特征值对边界和边值条件的单调依赖关系,本文建立了左定Sturm-Liouville与两个相关的定型Sturm-Liouville问题之间的特征值不等式关系.     

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.     

具有转移条件的两个区间Sturm-Liouville算子的Green函数和渐近展开(英文)

考虑具有转移条件的两个区间Sturm-Liouville算子.定义与转移条件相关联的的Hilbert空间,并在新Hilbert空间里讨论两个区间Sturm-Liouville算子.进一步构造具有转移条件的两个区间Sturm-Liouville算子的Green函数,并且给出两个区间Sturm-Liouville算子的特征值和特征函数的渐近展开.     

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.     

常型Sturm-Liouville问题的左定空间

本文刻画了常型Sturm-Liouville问题的左定空间的一般形式．根据自伴边值条件的分类,确切地给出了所有可能的左定空间描述．     

常型Sturm-Liouville问题的左定边值条件

本文刻画了常型Sturm-Liouville问题的左定边值条件．通过Sturm-Liouville微分算式的系数、区间端点以及边值条件给出了其左定性的充要条件．应用自伴边值条件分类,确切地给出了所有可能的左定边值条件．     

New canonical forms of self-adjoint boundary conditions for regular differential operators of order four

In this paper, we find new canonical forms of self-adjoint boundary conditions for regular differential operators of order two and four. In the second order case the new canonical form unifies the coupled and separated canonical forms which were known before. Our fourth order forms are similar to the new second order ones and also unify the coupled and separated forms. Canonical forms of self-adjoint boundary conditions are instrumental in the study of the dependence of eigenvalues on the boundary conditions and for their numerical computation. In the second order case this dependence is now well understood due to some surprisingly recent results given the long history and voluminous literature of Sturm-Liouville problems. And there is a robust code for their computation: SLEIGN2.     

The classification of self-adjoint boundary conditions: Separated, coupled, and mixed

There are three basic types of self-adjoint regular and singular boundary conditions: separated, coupled, and mixed. For even order problems with real coefficients, one regular endpoint and arbitrary deficiency index d, we give a construction for each type and determine the number of possible conditions of each type under the assumption that there are d linearly independent square-integrable solutions for some real value of the spectral parameter. In the separated case our construction yields non-real conditions for all orders greater than two. It is well known that no such conditions exist in the second order case. Our construction gives a direct alternative to the recent construction of Everitt and Markus which uses the theory of symplectic spaces. We believe our construction will prove useful in the spectral analysis of these operators and in obtaining canonical forms of self-adjoint boundary conditions. Such forms are known only in the second order, i.e. Sturm-Liouville, case. Even for regular problems of order four no such forms are available.     

The classification of self-adjoint boundary conditions of differential operators with two singular endpoints

For general even order linear ordinary differential equations with real coefficients and endpoints which are regular or singular and for arbitrary deficiency index d, the self-adjoint domains are determined by d linearly independent boundary conditions. These conditions are of three types: separated, coupled, and mixed. We give a construction for all conditions of each type and determine the number of conditions of each type possible for a given self-adjoint domain. Our construction gives a direct alternative to the recent construction of Everitt and Markus which uses the theory of symplectic spaces. We believe our construction will prove useful in the spectral analysis of these operators and in obtaining canonical forms of self-adjoint boundary conditions. Such forms are known only in the second order, i.e. Sturm-Liouville, case. Even for regular problems of order four no such forms are available. In the case when all d conditions are separated this construction yields explicit non-real conditions for all orders greater than two. It is well known that no such conditions exist in the second order case.     

Fliess series approach to the computation of the eigenvalues of fourth-order Sturm-Liouville problems

This paper is an extension to our work on the computation of the eigenvalues of regular fourth-order Sturm-Liouville problems using Fliess series. The purpose here is twofold. First, we consider general self-adjoint separated boundary conditions. Second, we modify the algorithm presented in an earlier paper to ease considerably the computation of the iterated integrals involved.     

High order approximations of the eigenvalues of sturm-liouville problems with coupled self-adjoint boundary conditions

Sampling theory has been used to compute with great accuracy the eigenvalues of regular and singular Sturm-Liouville problems of Bessel Type. We shall consider in this paper the case of general coupled real or complex self-adjoint boundary conditions. We shall present few examples to illustrate the power of the method and compare our results with the ones obtained using the well-known Sleign2 package.