Sturm’s 1836 Oscillation Results Evolution of the Theory
Sturm’s 1836 Oscillation Results Evolution of the Theory
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Sturm 的 1836 年振荡结果理论的演变
DOI：
10.1007/3-7643-7359-8_1
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发表时间：
2005
期刊：
影响因子：
--
通讯作者：
D. Hinton
中科院分区：
文献类型：
--
作者：
D. Hinton
我们研究如何斯特姆的振荡定理比较，分离，和索引的本征函数的零点的数量已经演变。这是Bocher谁第一次把证明严格的基础上，和主要的分析工具，介绍了皮科内，Prufer，莫尔斯，里德，和其他人。对于二阶情形，给出了一些基本的振动性和不共轭性结果。我们展示了如何定义的振动和不共轭有一个以上的解释高阶方程和系统，但它是从变分法的定义，提供了最富有成效的概念，他们也有应用的微分方程谱理论。系统的比较和分离定理，并显示它们如何适用于标量方程，给一个自然的延伸斯特姆的二阶情况。最后，我们回到二阶的情况下，显示如何索引的零特征函数的变化时，有一个参数的边界条件，或如果权函数改变的迹象。
We examine how Sturm’s oscillation theorems on comparison, separation, and indexing the number of zeros of eigenfunctions have evolved. It was Bocher who first put the proofs on a rigorous basis, and major tools of analysis where introduced by Picone, Prufer, Morse, Reid, and others. Some basic oscillation and disconjugacy results are given for the second-order case. We show how the definitions of oscillation and disconjugacy have more than one interpretation for higher-order equations and systems, but it is the definitions from the calculus of variations that provide the most fruitful concepts; they also have application to the spectral theory of differential equations. The comparison and separation theorems are given for systems, and it is shown how they apply to scalar equations to give a natural extension of Sturm’s second-order case. Finally we return to the second-order case to show how the indexing of zeros of eigenfunctions changes when there is a parameter in the boundary condition or if the weight function changes sign.