具有有限谱的微分方程边值问题的研究

项目名称： 具有有限谱的微分方程边值问题的研究

项目编号： No.11301259

项目类型： 青年科学基金项目

立项/批准年度： 2014

项目学科： 数理科学和化学

项目作者： 敖继军

作者单位： 内蒙古工业大学

项目金额： 22万元

中文摘要： 经典的Sturm-Liouville理论的结论是：对于正则的或奇异的自共轭的Sturm-Liouville问题，其谱是无界的，因而也是无穷的。但当此类问题的系数满足一定条件时，其谱却可以是有限的。之前，对于具有有限谱的微分方程边值问题的研究比较少，已有的结论也只是对二阶的Sturm-Liouville问题的研究。本课题拟深入研究具有有限谱的微分算子（也称微分方程边值问题）以及它与有限维矩阵特征值问题之间的关系，并进一步利用这种关系通过矩阵特征值问题的反谱理论给出对应具有有限谱的微分算子的反谱问题。我们对高阶的边值问题以及带有转移条件或边界条件中含有谱参数的边值问题进行研究，给出相应的有限谱结论及其与矩阵问题之间的关系；利用边值问题与矩阵特征值问题之间的关系给出对应具有有限谱的微分算子的反谱理论。上述结果将对深入研究两类问题以及无穷问题的有穷逼近起到重要作用。

中文关键词： 边值问题；有限谱；矩阵特征值问题；反谱问题；转移条件

英文摘要： The classical Sturm-Liouville theory state that the spectrum of a regular or singular, self-adjoint Sturm-Liouville problem is unbounded and therefore infinite. However,while the coefficients of the problem satisfying certain conditions,the spectrum of it may be finite. Previously, there is few studies on differential boundary value problems with finite spectrum, and the existing results are for the second order Sturm-Liouville problems only.In this program we will systematically investigate the differential operators(or differential boundary value problems) with finite spectrum and the relationship between these problems and the finite dimensional matrix eigenvalue problems. By using the given relationship above mentioned, we show the corresponding inverse spectral problems of the differential operators with finite spectrum by inverse matrix eigenvalue problems. We study higher order problems and Sturm-Liouville problems with transmission conditions and\or eigenparameter-dependent boundary conditions, show the finite spectrum results and their matrix representations; We also give the inverse spectral theory of differential operators with finite spectrum by using the relationship of the two problems. It is of great significance to both the differential boundary value problems and matrix eigenvalue problems, and

英文关键词： boundary value problem；finite spectrum；matrix eigenvalue problem；inverse spectral problem；transmission condition